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+{"text":"\\section{introduction}\n\\label{introduction}\nBy far the most acute and clear-cut tests of MOND \\cite{milgrom83a} come from the dynamical analysis of rotation curves (RCs) of disc galaxies. For reviews of MOND see, e.g. Refs. \\cite{fm12,milgrom14}.\n\\par\nIn particular, the MOND acceleration constant, $\\az$, appears in such analyses in different roles: as the boundary constant marking the transition from Newtonian behavior to deep-MOND, as setting behavior in the deep MOND regime, for example, fixing the normalization of the MOND mass-asymptotic-speed relation (MASR) -- underlying the baryonic Tully-Fisher relation (BTFR), and in dictating the RCs of low-acceleration (or low-surface-brightness) galaxies. It appears in several roles in the much discussed MOND prediction \\cite{milgrom83a} of the mass-discrepancy-acceleration relation (for tests of this prediction see, e.g., \\cite{sanders90,mcgaugh04,tiret09,wu15,mcgaugh16a}). It also appears in the no-less-striking central-surface-densities relation, which is different and independent of the other MOND relations \\cite{milgrom09a,lelli16,milgrom16a}.\n\\par\nThe value that has emerged for $\\az\\approx 1.2\\times 10^{-8}\\cmss$, as been recognized early on (Milgrom 1983a) to have cosmological connotations. In particular we have:\n\\beq \\bar\\az\\equiv 2\\pi \\az\\approx cH_0\\approx c^2(\\Lambda\/3)^{1\/2}, \\eeqno{coinc}\nwhere $H_0$ is the Hubble constant, and $\\Lambda$ the observed equivalent of a cosmological constant.\n\\par\nThe Former of these near equalities, and the realization that MOND may well be an effective theory rooted somehow in cosmology, have pointed to the possibility that $\\az$, or some aspects of MOND, may be varying with cosmological time  so as to retain the first equality at all times.\n\nThe obvious way to test this possibility, given that $\\az$ is sharply determined by rotation-curve analysis, is to analyze RCs of high-$z$ galaxies to see if their dynamics can be accounted for by MOND, and whether this requires $\\az$ to be $z$ dependent (an early attempt at this is described in Ref. \\cite{milgrom08}).\n\\par\nIn recent years, there have been several studies of the internal kinematics of high-$z$ galaxies (e.g., \\cite{price16,wuyts16,lang17}). These are, by and large, statistical in nature.\n\\par\nGenzel et al. \\cite{genzel17} have recently published the individual RCs of six high-redshift galaxies\n($z\\sim 0.9-2.4$), and have presented a thorough dynamical analysis of them.\nThese are selected from a large sample of several hundred, according to criteria that are conducive to cleaner analysis.\nThis sample now also affords a closer, if preliminary, examination of the dynamics of high-$z$ in light of MOND.\n\\par\nThe main general conclusions of Ref. \\cite{genzel17} are that these galaxies are `baryon dominated' within the studies radii, and that they show marked decline in the RC still within the optical image.\nIn both regards, this is very reminiscent of the findings of Ref. \\cite{romanowsky03} of `dearth of dark matter in ordinary elliptical galaxies' (at low redshift) -- based on planetary-nebulae velocities.\n\\par\nI shall show that both of these characteristics of the high-$z$ disc galaxies of Ref. \\cite{genzel17} follow from MOND because these galaxies have accelerations within the studied regions that are higher than $\\az$. The MOND analysis by Ref. \\cite{ms03} and Ref. \\cite{tian16} showed this for low-$z$, elliptical galaxies.\n\\par\nIt is important to keep in mind that for natural reasons the data of Ref. \\cite{genzel17} are not up to the standard afforded by RCs and baryon distributions available for dynamical analysis in the nearby Universe. In comparison with the latter they are limited in scope, and they are subject to large uncertainties (partly reflected in their large quoted errors).\nSome concerns that come to mind are: a. The inclinations of the six galaxies are $i({\\rm deg})=75\\pm 5,~30\\pm 5,~62\\pm 5,~25\\pm 12,~45\\pm 10,~34\\pm 5$. Three of them have low inclinations $i< 35$ degrees. Such low inclinations are generally considered problematic because it is difficult to measure such low inclinations accurately, and because the actual rotational speeds, and the acceleration deduced from them are sensitive to the exact value (the accelerations scaling as $1\/sin^2 i$).\nb. This is further compounded by the fact that\nthese RCs are not based on 2-D tilted-ring derivation as the standard has come to require, and hence do not account for possible variable position angle and inclination, especially problematic for low-inclination galaxies.\nc. Large random motions are present in these galaxies; so large (and uncertain) asymmetric-drift corrections have to be applied. d. The luminosity distribution is measured in the rest-frame optical-band, not as good for converting light to mass compared with far IR now used routinely for local galaxies. e. Some of the galaxies have a substantial bulge, and the necessary separation to components, with possibly different M\/L values, is problematic.\nf. Kinematics are measured from $H_\\a$ velocities, so are confined to the optical image with no analog of the extended HI RCs.\\footnote{In many of these regards the quality of these RCs may be likened to that of RCs available in the late 1970s for low-$z$ galaxies, before extended HI RCs became available.}\n\\par\nStill, these are the best RC data we have at present for such high redshift, and thus are valuable in constraining cosmological evolution of either DM scenarios, or, as here, $z$-dependence of MOND.\n\\par\nIn Sec. \\ref{MOND}, I give some MOND formulae needed here. Section \\ref{results}\ncompare the MOND predictions with the results of Ref. \\cite{genzel17},\nand Sec. \\ref{discussion} is a discussion.\n\n\\section{Relevant MOND formulae}\n\\label{MOND}\nIf at some radius, $R$, in the midplane of a disc galaxy $\\gN(R)$ is the Newtonian acceleration calculated from the baryon distribution, and $g$ is the dynamically determined acceleration, then MOND predicts the relation \\cite{milgrom83a}:\n\\beq g\\m(g\/\\az)=\\gN, \\eeqno{mdarmu}\nwhere $\\az$ is the MOND acceleration constant, and $\\m(x)$ the `interpolating function' for rotation curves.\nThis can be written equivalently in terms of the $\\n(y)$ interpolating function\nas\n\\beq g=\\gN\\n(\\gN\/\\az), \\eeqno{mdarnu}\nwhere $\\n(y)$ is related to $\\m(x)$  by $\\m(x)=1\/\\n(y)$, where $x=y\\n(y)$ [and so $y=x\\m(x)$].\nThese equivalent forms are known as the mass-discrepancy-acceleration relation (MDAR) since\n\\beq \\eta\\equiv g\/\\gN=1\/\\m(g\/\\az)=\\n(\\gN\/\\az) \\eeqno{mdar}\ncan be identified as the mass discrepancy.\n\\par\nAs convention goes, Ref. \\cite{genzel17} define the dark-matter fraction at $R$ -- better referred to in the present context as the `phantom-matter' fraction --as\n$\\zeta(R)=[V\\_{DM}(R)\/V(R)]^2$, or in terms of the accelerations\n\\beq \\zeta(R)\\equiv (g-\\gN)\/g. \\eeqno{frac}\nThus, MOND predicts\n\\beq \\zeta=1-\\eta^{-1}=1-\\m=1-1\/\\n.  \\eeqno{etamu}\n\nI will show the MOND predictions for $\\zeta$ for two forms of the MOND interpolating function used routinely for RC fits and fits to the MDAR.\nThe first is\n\\beq \\m(x)=\\frac{x}{1+x}, \\eeqno{i}\nwhich gives a `phantom-matter' fraction of $\\zeta=(1+x)^{-1}$.\nThe other takes a simple form in the $\\n(y)$ language \\cite{ms08,mcgaugh08,mls16,milgrom16}:\n\\beq \\n(y)=(1-e^{-\\sqrt{y}})^{-1},   \\eeqno{ii}\nwhich gives $\\zeta=e^{-\\sqrt{y}}$.\nThese two interpolating functions differ by at most $\\sim 5\\%$ over the full range of arguments and so predict almost indistinguishable rotation curves.\nHowever, in the region of high accelerations, where both functions are nearly 1, they differ substantially in their exact departure from 1. So, when the predicted\nfractions of `phantom matter' are small, the two functions can give different predictions for this small quantity.\n\\par\nAnother MOND prediction I will need: Given the total baryonic mass, $M_b$, MOND predicts \\cite{milgrom83a,milgrom83b} for an isolated galaxy, an asymptotically flat RC, with the constant rotation speed\n\\beq \\vinf^4=M_bG\\az.  \\eeqno{iii}\nThis is the MASR mentioned in Sec. \\ref{introduction}\n\\section{Results}\n\\begin{table*}[t]\n \\begin{tabular}{lccccccccl}\n \\hline\\hline\n\n\nGalaxy& $z$ & $\\raf$ & $V_c(\\raf)$ & $M_b$& $\\zef$& $\\vinf$ & $x\\_{1\/2}$& $\\zeta\\^M\\_{1\/2,a}$ & $\\zeta\\^M\\_{1\/2,b}$  \\\\\n& & $\\kpc$ & $\\kms$ & $10^{11}\\msun$ & & $\\kms$&&&\\\\\n\\hline\nCOS4 01351   &  0.854 & 7.3  & 276 & 1.7 & $0.21 (\\pm0.1$) & 228 & 2.8 &0.26& 0.22 \\\\\nD3a 6397 & 1.500&  7.4      & 310 & 2.3  & 0.17 ($<$0.38)\t& 246 & 3.5 &0.22& 0.18 \\\\\nGS4 43501  & 1.613 &  4.9  & 257 & 1.0 & $0.19 (\\pm0.09)$ & 200 & 3.6 & 0.22 & 0.17 \\\\\nzC 406690 &2.196\t&  5.5   & 301  &  1.7  & $0 (<0.08)$ & 228 & 4.4 & 0.18 & 0.14 \\\\\nzC 400569 & 2.242  &  3.3      & 364    \t & 1.7 & $0 (<0.07)$ & 228 & 10.8 & 0.08 & 0.04 \\\\\nD3a 15504 & 2.383  &  6      & 299    \t & 2.1 & $0.12 (<0.26)$ & 240 & 4.0 & 0.20& 0.16\\\\\n\\hline\n\\end{tabular}\n\\caption{Galaxy name \\{column 1\\}, its redshift \\{2\\}. Columns 3-6 are best-fit attributes deduced by Ref. \\cite{genzel17}: the half-light radius, $\\raf$, (in the rest-frame optical band) \\{3\\}; the rotational speed there (corrected for inclination and asymmetric drift) \\{4\\}; the total baryonic mass \\{5\\}; and the dark-matter fraction, $\\zef$ at $\\raf$, with errors or upper limits \\{6\\}. Column 7-10 show calculated MOND quantities: the predicted asymptotic rotational speed, $\\vinf$, based on $M_b$, from eq.(\\ref{iii}) \\{7\\}, The acceleration at $\\raf$ in units of $\\az$ \\{8\\}, the expected MOND value of $\\zef$ based on the interpolating function of eq. (\\ref{i}), $\\zeta\\^M\\_{1\/2,a}$ \\{9\\}, and that based on eq. (\\ref{ii}), $\\zeta\\^M\\_{1\/2,b}$ \\{10\\}, all calculated for the nearby-Universe value of $\\az=1.2\\times 10^{-8}\\cmss$.}\n\\label{table}\n\\end{table*}\n\\label{results}\nTable \\ref{table} shows the values of the relevant parameters as they appear in Table 1 of Ref. \\cite{genzel17}. I show in the table, and use, the relevant quantities given in Ref. \\cite{genzel17} as their best fit model parameters (resulting from fitting the rotation curves to mass models that include baryons and dark matter): the half light radius, $\\raf$, the dynamical rotational speed at $\\raf$, and the total baryonic mass, $M_b$. For $M_b$ and $\\raf$ they also give their pre-fit, directly estimated values. In most cases, the former values, which I use, lie within the error range of the latter. I also show their deduced values of the `phantom-matter' fractions, $\\zef$, at $\\raf$, and the values of $\\zef$ predicted by MOND for the two commonly used interpolating functions, all as detailed in Sec. \\ref{MOND}.\n\\par\nWe see that as found by Ref. \\cite{genzel17} the MOND $\\zef$ values are small -- a few tens of percents at most. Furthermore, except for the rogue zC 406690, where the upper limit is lower than my estimates, the MOND predictions are, case by case, in good agreement with what Ref. \\cite{genzel17} give.\\footnote{But beware that the $\\zef$ values of Ref. \\cite{genzel17} are based on model best-fits with NFW dark-matter distributions. Given their large uncertainties on $M_b$, their RCs are probably also consistent with sub-maximal discs, and rather larger values of $\\zef$. The distinction between maximal and sub-maximal discs is moot even with much better data.\\label{fnq}} And note that zC 406690 has a quoted inclination of $i=25\\pm 12$ degrees; so it's kinematic analysis is practically useless.\n\n\n\n\n\\subsection{Falling rotation curves}\nThe RCs shown by Ref. \\cite{genzel17} exhibit some decline beyond their maximum. Such decline is also typical of high-surface-brightness galaxies in the local Universe (see, e.g. some early-type galaxies in the sample of Ref. \\cite{sn07}, in particular, their RC for UGC 4458, which drops from $\\sim 500\\kms$ to $\\sim 300\\kms$ within $10 \\kpc$ and then becomes flat at $\\sim 250\\kms$ to $55\\kpc$).\n\\par\nSuch a decline seems to be more prevalent in the high-$z$ samples at hand (see also Ref. \\cite{lang17}). Part of the reason, as extensively discussed by Refs. \\cite{genzel17,lang17}, is that rather more than in low-$z$ galaxies, velocity dispersions in the disc contribute substantially to the balance against gravity, hence diminishing the role of rotational support. It is notoriously difficult and uncertain to correct for this important effect. Indeed, in the stacked RCs of Ref. \\cite{lang17} (see their Fig. 8), galaxies with\nhigh rotation-to-dispersion ratio show much less marked decline than those with small values.\n\\par\nOne should also consider the effects of selection: High-surface-brightness galaxies -- where such declines are also observed at low-$z$ -- are naturally more amenable to measurements at high redshift, and are easier to follow to larger radii. Indeed, Fig. 5 of Ref. \\cite{lang17} shows that the number of galaxies contributing at the outer radii, where the decline is evident, is much smaller than the total in the sample: $\\sim 12$ galaxies that contribute down to the outer stacked-data point, compared with $\\sim 90$ that contribute at low radii. These may well be selecting preferentially higher-surface-brightness galaxies.\n\\par\nIn MOND, we do expect marked decline beyond the maximum in galaxies with mean accelerations that are so high compared with $\\az$. For example, MOND-predicted rotation curves of such model galaxies are shown in Figs. 1 and 2\nof Ref. \\cite{milgrom83b} (the models with high $\\xi\\sim 5$ there). And see also Fig. 2 of Ref. \\cite{ms03} for the predicted MOND RC of the elliptical NGC 3379, which drops from $\\sim 300\\kms$ at maximum to $\\sim 200 \\kms$.\n\\par\nWe can estimate the room for a drop in the velocity allowed by MOND for the six galaxies under study, by comparing the observed maximum speed with the predicted asymptotic rotational speed, $\\vinf$, which can be deduced from the estimates of the baryonic masses, using eq. (\\ref{iii}) -- assuming that the galaxy is isolated. These estimates are given in Table \\ref{table} based on the best-fit values that Ref. \\cite{genzel17} give for $M_b$. Note that the direct estimates of $M_b$ given by Ref. \\cite{genzel17} have large quoted errors given in all cases as $\\pm 50\\%$ (i.e., a factor of $\\sim 3$ in range), corresponding to a relative error of $+0.1~-0.15$ in $\\vinf$. From Fig. 2 of Ref. \\cite{genzel17} one sees that the maximum speed for the galaxies is about\n$1.1V(\\raf)$, and occurs at $\\sim 1.5 \\raf$. We see then that the estimated ratio $\\vinf\/V_{max}$ is as low as $\\sim 0.55~ (\\pm 0.1)$ (for one of the 6 galaxies, zC 400569), and is $\\sim 0.7 (\\pm 0.1)$ for most others. This would allow the drops Ref. \\cite{genzel17} estimate (these are subject to substantial uncertainties due to the uncertain asymmetric-drift correction, and possible unaccounted for warps). For one galaxy, zC 406690, Ref. \\cite{genzel17} estimate a very large drop. But, as I pointed out above, this is quite unreliable as the stated inclination for this galaxy is $i=25\\pm 12$ degrees.\n\\par\nIn MOND, the presence of neighboring bodies can also contribute to the decline of the RCs through the external-field effect (e.g., Refs. \\cite{milgrom83a,wu15,haghi16,mcgaugh16}. According to Ref. \\cite{genzel17} their 6 galaxies are relatively isolated, so this should not be a factor, but it is hard to asses the importance of the effect in statistical studies such as that of Ref. \\cite{lang17}.\n\n\n\\section{Discussion}\n\\label{discussion}\nThe results of Ref. \\cite{genzel17} are well accounted for by MOND in the very form that has been applied successfully to low-$z$ galaxies, with the canonical value of $\\az$.\n\\par\nAlthough these RCs do not probe the deep MOND regime -- where MOND enters in full glory -- they do vindicate an important prediction of MOND that does not arise naturally in the dark-matter paradigm. Namely, that mass anomalies should be small (sub-dominance of `phantom matter') at accelerations above $\\az$.\nThat this is now seen to be the case also at high $z$ even sharpens the case for MOND: It shows this prediction to be independent of the evolutionary status of the galaxies, strengthening the case for a law of nature as the origin, rather than some complicated and contrived evolutionary processes.\n\\par\nIt appears that these results cannot accommodate much higher values of $\\az$ at high redshift. Looking at Table \\ref{table}, we see that, for example, a value of the MOND acceleration of $4\\az$ would have resulted in $x\\_{1\/2}$ values for the higher-$z$ galaxies of order 1. This would have predicted $\\zef$ values of order 0.5, which would be uncomfortably in tension with the values estimated by Ref. \\cite{genzel17}\\footnote{Values of the MOND constant smaller than $\\az$ cannot be excluded, but they are anyhow less motivated.} (But, remember footnote \\ref{fnq}.) This constraint makes use, essentially of the role of $\\az$ in MOND as `boundary acceleration'. Another, independent constraint is based on the role of $\\az$ as setting the MASR normalization: With a value of the MOND constant as high as $4\\az$ the predicted values of $\\vinf$ in  Table \\ref{table} should be increased by a factor of $4^{1\/4}\\sim 1.4$, making $\\vinf\/V_{max}\\sim 1$, not leaving room for decline beyond the maximum, unless the baryonic masses are substantially lower.\n\\par\nIdeally, we could test for variations of $\\az$ by searching for evolution in the proportionality constant of the MASR, eq. (\\ref{iii}). But this is not possible with the present data, as clearly they do not reach the asymptotic speeds, as required by the MOND MASR.  `Evolution' of the zero point of some versions of the BTFR, using available velocity measures such as the maximum speed have been studied. But these are not what the MOND MASR dictates, and cannot be used to constrain cosmological variations of the MOND constant.\nIt is an opportunity to stress again the distinction between various versions of the BTFR, and the specific version MOND predicts as the MASR, which employs the asymptotic speed.\n\\par\nThis result may help constrain ideas that rest on the MOND constant varying with cosmic time, such as the suggestion that the first of the near equalities in eq. (\\ref{coinc}) held at all times, or other possible variations (see discussion in Ref. \\cite{milgrom09} and references therein). Ref. \\cite{milgrom99} offers a possible causal connection between $\\az$ and $\\Lambda$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction} \nBecause of the dominance of lines of atomic iron in the spectra of cool stars, the iron abundance is often used as a proxy for total metal content, or metallicity. Neutral and singly ionised iron with different properties are also frequently used for determination of spectroscopic stellar parameters. This makes [Fe\/H] arguably the most important abundance indicator when studying the evolution of stars and galaxies. The iron abundance of the Sun itself is important not only as an anchor for the cosmic [Fe\/H]-scale, but it also influences the structure and evolution of stars because it is a large opacity contributor \\citep[][and references therein]{Bailey15}. \n\nIt has long been known that the ionisation balance of FeI--FeII departs from local thermodynamic equilibrium (LTE) in the photospheres of late-type stars \\citep[e.g.][]{Athay72,Rutten84,Thevenin99, Korn03}. However, non-LTE (hereafter NLTE) calculations of neutral iron have suffered from large systematic uncertainties due to poorly constrained atomic data, in particular the efficiency of collisions with electrons and neutral hydrogen \\citep[e.g.][]{Mashonkina11b}. Further, the large complexity of the atomic structure of iron has prevented consistent NLTE calculations to be performed with realistic 3D radiation-hydrodynamical simulations of solar and stellar photospheres. \n\nIn Paper I and II of this publication series \\citep{Bergemann12,Lind12a}, we presented a model atom for iron with the efficiency of H collisions calibrated using high-quality spectroscopic observations of well-studied benchmark stars, including the Sun. We employed both standard 1D atmospheric models and so called average-3D models (hereafter $\\rm\\langle3D\\rangle$), which are spatial and temporal averages of full 3D radiation-hydrodynamical simulations of stellar atmospheres. In Paper III \\citep{Amarsi16b} we presented a new model atom including quantum mechanical calculations of hydrogen collisions (Barklem, in prep.) and demonstrated its performance for metal-poor benchmark stars using full 3D NLTE calculations. Here we further improve the atom and confront our 3D NLTE predictions with the observed centre-to-limb variation of iron lines in Sun. \n\n\\citet{Nordlund84,Nordlund85} pioneered the investigation of NLTE line formation of iron in 3D hydrodynamical model atmospheres more than three decades ago. The first paper studied the departure of FeI--FeII from Saha ionisation balance and reported significant $\\rm(0.2\\,dex)$ over-ionisation of the neutral species. The second paper used a two-level FeI atom, coupled to a FeII continuum, and predicted significant line weakening of the example FeI line at 5225\\AA\\  due to a superthermal source function. \n\n\\citet{Shchukina01} later studied NLTE line formation in a hydrodynamical model of the Sun in the so called 1.5D approximation, neglecting horizontal radiative transfer. They used a 248 level FeI+FeII atom and concluded that NLTE effects vary strongly with the granulation pattern and the FeI line properties, with a net NLTE correction to FeI line abundances of up to $+0.12$\\,dex for the lowest-excitation lines. The only previous work investigating NLTE line formation of iron in the Sun using a multi-level atom and full 3D radiative transfer is the series by \\citet{Holzreuter12,Holzreuter13,Holzreuter15}, in which the authors rigorously compare synthetic line profiles generated under different assumptions. However, they were limited to using a strongly simplified 23-level atom and made no quantitative comparison to observations. These earlier studies have in common that they included only experimentally known energy levels of iron and neglected the influence of hydrogen collisions on the statistical equilibrium, both of which exaggerate the NLTE effects.  \n\nWe present full 3D NLTE calculations using a comprehensive 463 level atom with realistic atomic data to enable a direct comparison to the most constraining observations possible, i.e. high-spectral resolution and high signal-to-noise (S\/N) observations of the Sun at different viewing angles. The paper is divided in the following sections: Sect.\\,\\ref{sect:method} outlines the observations, the assembly and reduction of the model atom, and the method used for spectral synthesis.  Sect.\\,\\ref{sect:results} presents the results for the solar centre-to-limb variation of iron lines and the solar iron abundance. Sect.\\,\\ref{sect:conc} summarises our conclusions.  \n\n\\section{Method}\n\\label{sect:method}\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.43,viewport=3cm 1cm 21cm 21cm]{Positions}\n\\caption[]{Overview of the SST pointings on the solar disk, inclined by the heliographic latitude of the observer.  The blue circles mark the targeted $\\mu$-angles and $\\mu=0.999$ for reference.} \n\\label{fig:pos} \n\\end{center} \n\\end{figure}\n\n\n\\subsection{Observations}  \nWe acquired spectroscopic data with high spatial and spectral resolution using the TRIPPEL \\citep{Kiselman11} instrument at the Swedish 1-m Solar Telescope \\citep[SST,][]{Scharmer03} on La Palma. The observing campaign lasted from 23 June to 8 July, 2011. \n\nThree spectrographic cameras and three imaging cameras were operated simultaneously.  Three  different setups were used, resulting in a total of nine spectral windows with wavelength bands specified in Table \\ref{tab:bands}. Two slit-jaw cameras recorded simultaneous images at approximately 5320\\AA\\, and 6940\\AA\\,, respectively. The third camera was used to monitor the magnetic activity of the region with a 1.1\\AA\\ filter centered on the Ca\\,II H line. Five different heliocentric angles on the solar disk were targeted, corresponding to $\\mu\\equiv\\cos{\\theta}=0.2, 0.4, 0.6, 0.8$ and $1.0$, where $\\theta$ is the angle between the ray direction and the surface normal. The number of observations at each pointing is listed in Table \\ref{tab:bands}, discarding exposures that failed due to suspected tracking problems (usually due to very bad seeing), or regions with obvious activity as deemed from the Ca\\,II~H core emission.\n\nThe intensity contrast peaks at disk centre and exposures were made while scanning the spectroscopic slit over a small region in order to reduce the imprint of the local granulation pattern. At other pointings, the slit position was held fixed and aligned parallel to the closest part of the solar limb. The telescope field rotation caused the actual position selected in this way to depend on the time of day, as is evident in Fig.~\\ref{fig:pos}. Of the two possible choices for a specific $\\mu$ value and time of day, the one showing the least activity was preferred. \n\nFor the $\\mu=0.2$ pointings, the position of the slit could be measured accurately using the slit-jaw images, which include the solar limb. For the other pointings, $\\mu$ was determined from the output of the telescope tracking system. In order to get the readings as accurate as possible, frequent calibrations by pointing at the limb at four position angles to find the solar centre were made. Somewhat conservatively, we estimate the accuracy in the the nominal values to be $\\pm 15\\hbox{$^{\\prime\\prime}$}$.\n\nAs evident from Fig. \\ref{fig:pos} and Table \\ref{tab:bands}, the pointing accuracy as expressed in $\\mu$ degrades with decreasing $\\mu$ value, i.e. from the centre towards the limb. The $\\mu=0.2$ pointing deviates from this trend since it was determined from the slit-jaw images and the largest uncertainties thus affect the $\\mu=0.4$ observations. In total we kept 147 pointings, 4-20 for each $\\mu$-value and configuration, as detailed in Table \\ref{tab:bands}. For each $\\mu$ pointing and wavelength, Table~\\ref{tab:bands} gives the mean value of $\\mu$ and its error estimate calculated from the standard deviation of the nominal position readings combined with a systematic error. For $\\mu\\ge0.4$ the systematic component was computed using the $15\\hbox{$^{\\prime\\prime}$}$ calibration error and for $\\mu=0.2$, we used the approximate spatial extent of the slit in the same way as \\citet{Pereira09a}. \n\nThe data were reduced using the same method and software as described in \\citet{Pereira09a}. First, the data were corrected for dark current and flat fielded using calibration exposures taken in close connection to the observations. \nGeometrical distortions in the spectrograms were then removed using polynomial fits to the location of selected spectral lines (for smile) and with the help of a grid place across the slit (for keystone). Wavelength calibration was made by cross-correlation of spectral lines with the disk-centre FTS atlas of Brault \\& Noyes (1987)\\footnote{\\url{ftp:\/\/ftp.hs.uni- hamburg.de\/pub\/outgoing\/FTS- Atlas}}. The same atlas was used to model the internal straylight of the spectrograph (assumed constant over each spectrogram) as well as systematic spectral artefacts that the flat-fielding cannot correct for. The result of the reduction procedure is to force mean spectra from the quiet disk centre to be as close as possible to the reference atlas spectrum. The same corrections are then applied to all spectra.\n\nIn this paper, we analyse the centre-to-limb behaviour of iron lines and create a single 1D spectrum for each pointing by coadding the individual spectrograms and then forming an average along the slit direction. This increases the S\/N and, following \\citet{Pereira09b}, removes the need for Fourier filtering of photon noise that was applied by \\citet{Pereira09a}. The S\/N per pixel of the average spectra ranges between 1000--4000 at a spectral resolution of $\\lambda\/\\delta\\lambda\\approx150,000$. The S\/N ratio was estimated from the median standard deviation of all measurements at a given wavelength.\n\n\\begin{table*}\n      \\caption{Summary of the observational configuration. Columns A-C give the wavelength band of each of the three spectrographic cameras. $\\#$ represents the number of pointings.}\n         \\label{tab:bands}\n         \\centering\n         \\begin{tabular}{llllllllllllll}\n                \\hline\\hline\n                 Set & A & B & C &  $\\#$ & $\\mu$ &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$ \\\\\n                 & [\\AA\\ ]& [\\AA\\ ]& [\\AA\\ ]   \\\\\n                                \\hline\n\t      1\t& 5366-5377 &  6147-6159 & 8710-8728 & 6 \t& 0.201 \t\t& 4 \t\t& 0.380\t      \t& 7 \t& 0.600     \t& 7 \t&  0.802 \t\t& 4 \t\t& 1.0000 \t\t\t\\\\%\t      1 &  537 &  615 & 873 \\\\\n\t         &     \t\t      &                       &                      &\t& $\\pm0.007$ \t&   \t\t& $\\pm0.032$\t&  \t& $\\pm0.014$\t&  \t& $\\pm0.005$ \t&  \t\t& $\\pm0.0005$\t\t\\\\\n\t      2 & 5378-5390 &  6159-6172 & 8727-8744 & 7 \t& 0.205\t\t& 7 \t\t& 0.393\t  \t& 7 \t& 0.604\t   \t& 6 \t&  0.803\t \t& 12 \t\t& 1.0000\t \t\t\\\\%\t      2 &  538 &  616 & 874 \\\\\n\t         &     \t\t      &                       &                      &\t& $\\pm0.005$ \t&   \t\t& $\\pm0.019$\t&  \t& $\\pm0.009$\t&  \t& $\\pm0.005$ \t&  \t\t& $\\pm0.0003$\t\t\\\\\n\t      3 & 8656-8668 &  7825-7842 & 8691-8708 & 16\t& 0.203\t \t& 16\t\t& 0.397\t\t& 19 & 0.603\t  \t& 20 \t&  0.801\t \t& 9 \t\t& 1.0000\t \t\t\\\\\n\t      \t &     \t\t      &                       &                      &\t& $\\pm0.006$ \t&   \t\t& $\\pm0.027$\t&  \t& $\\pm0.006$\t&  \t& $\\pm0.004$ \t&  \t\t& $\\pm0.0005$\t\t\\\\\t      \t      \n                   \\hline \n         \\end{tabular}\n\\end{table*} \n\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.33,viewport=2cm 1cm 25cm 21cm]{LabK14}\n\\includegraphics[scale=0.33,viewport=2cm 0cm 25cm 21cm]{LabK13}\n\\caption[]{Comparison between theoretically predicted \\citep{K13,K14} and experimentally measured (see text for references) oscillator strengths of FeI and FeII. The experimental uncertainties are plotted as error bars. For FeII, a representative experimental uncertainly was set to $0.05$\\,dex.} \n\\label{fig:fvalues} \n\\end{center} \n\\end{figure}\n\n\n\\subsection{Atomic data} \n\\label{sect:atomdata}\n\nThe non-LTE calculations are performed by iterative solutions of the radiative transfer and statistical equilibrium equations, until the level populations have converged at all points in the atmosphere. The statistical equilibrium solution requires knowledge of the relevant radiative and collisional transition probabilities, which are collected in a model atom. The literature sources and databases used to assemble the model atom were listed in \\cite{Amarsi16b}. In this section we reiterate the main points and present more detail. \n\n\\begin{table*}\n      \\caption{Atomic data for the iron lines used for the centre-to-limb analysis. The $W_\\lambda$ columns list the equivalent widths measured for the five different $\\mu$-angles using direct integration over the wavelength range specified by $\\lambda_{\\rm int}$.}\n         \\label{tab:lines}\n         \\centering\n         \\begin{tabular}{lllllllllllllc}\n                \\hline\\hline\n\t\tIon & $\\lambda_{\\rm air}$ & $E_{\\rm low}$ & $\\log(gf)$ & $\\log(\\gamma)$ & $\\sigma^{(b)}$& $\\alpha^{(b)}$& $C_4^{(c)}$ & \\multicolumn{5}{c}{$W_{\\lambda}$ [m\\AA ]}   &$\\lambda_{\\rm int.}$\\\\\n                          & [\\AA\\ ]          & [eV]                   &                   &   Rad.\\,$^{(a)}$   &                   &                          &                        &    $\\mu=1.0$  &    $\\mu=0.8$  &    $\\mu=0.6$  &    $\\mu=0.4$  &    $\\mu=0.2$  &  [\\AA ] \\\\\n                                \\hline\nFeI & 5367.4659  \t& 4.415\t&   0.443$^{(d)}$   \t& 8.32 \t&    972 & 0.280\t&  -13.11 &   168.5  &   166.3   &  164.4  &   158.7   &  143.4   & $5367.10-5368.10$ \\\\\n&      &            \t\t&     \t\t&                \t\t\t& \t\t&    \t&         &  $\\pm1.3$  & $\\pm1.3$  & $\\pm1.4$  & $\\pm1.2$  & $\\pm1.5$ \\\\\nFeI & 5373.7086  & 4.473&  -0.710$^{(e)}$   & 8.13 &   1044 & 0.282  &  -13.76 &    59.4  &    58.4   &   58.5  &    58.5   &   54.6   & $5373.62-5373.82$ \\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.4$  & $\\pm0.6$  & $\\pm0.6$  & $\\pm0.7$ \\\\\nFeI & 5379.5736  & 3.695&  -1.514$^{(d)}$   & 7.85 &    363 & 0.249  &  -15.51 &    62.8  &    63.1   &   64.2  &    66.4   &   66.1   & $5379.20-5379.72$\\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.7$  & $\\pm0.6$  & $\\pm0.9$  & $\\pm0.8$ \\\\\nFeI & 5383.3685  & 4.313&   0.645$^{(d)}$   & 8.30 &    836& 0.278  &  -13.83 &   219.3  &   217.7   &  214.9  &   208.3   &  188.4   & $5382.70-5384.00$\\\\\n &   &           &     &                & &            &         &  $\\pm1.8$  & $\\pm1.9$  & $\\pm1.6$  & $\\pm1.4$  & $\\pm1.6$ \\\\\nFeI & 5386.3331  & 4.154&  -1.670$^{(f)}$   & 8.45 &    930& 0.278  &  -13.02 &    31.0  &    31.9   &   33.1  &    34.7   &   34.8   & $5386.10-5386.45$\\\\\n &   &           &     &                & &            &         &  $\\pm0.3$  & $\\pm0.4$  & $\\pm0.3$  & $\\pm0.5$  & $\\pm0.4$ \\\\\nFeI & 5389.4788  & 4.415&  -0.418$^{(g)}$   & 8.32 &    959& 0.280  &  -13.53 &    87.9  &    86.8   &   86.2  &    85.4   &   80.4   & $5389.30-5389.65$\\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.6$  & $\\pm0.7$  & $\\pm0.9$  & $\\pm0.7$ \\\\\nFeII& 6149.2459  & 3.889&  -2.840$^{(h)}$   & 8.50 &    186& 0.269  &  -16.11 &    38.3  &    38.4   &   37.2  &    36.7   &   31.4   & $6149.05-6149.40$\\\\\n &   &           &     &                & &            &         &  $\\pm0.3$  & $\\pm0.5$  & $\\pm0.6$  & $\\pm0.6$  & $\\pm0.5$ \\\\\nFeI & 6151.6173  & 2.176&  -3.299$^{(i)}$   & 8.29 &    277& 0.263  &  -15.55 &    49.0  &    49.6   &   52.2  &    55.4   &   55.8   &$6151.30-6151.85$ \\\\\n &   &           &     &                & &            &         &  $\\pm0.8$  & $\\pm0.7$  & $\\pm0.5$  & $\\pm0.7$  & $\\pm1.6$ \\\\\nFeI & 6157.7279  & 4.076&  -1.160$^{(f)}$   & 7.89 &    375& 0.255  &  -15.36 &    62.6  &    61.6   &   62.4  &    63.2   &   60.2   & $6157.50-6157.85$\\\\\n &   &           &     &                & &            &         &  $\\pm0.7$  & $\\pm0.6$  & $\\pm0.7$  & $\\pm0.9$  & $\\pm1.2$ \\\\\nFeI & 6165.3598  & 4.143&  -1.473$^{(d)}$   & 8.00 &    380& 0.250  &  -15.34 &    44.5  &    44.8   &   45.2  &    45.9   &   44.6   & $6165.25-6165.55$\\\\\n &   &           &     &                & &            &         &  $\\pm0.4$  & $\\pm0.5$  & $\\pm0.4$  & $\\pm0.6$  & $\\pm0.6$ \\\\\nFeI & 8699.4540  & 4.956&  -0.370$^{(e)}$   & 8.74 &    817& 0.272  &  -14.59 &    73.7  &    72.4   &   70.7  &    68.0   &   61.9   & $8699.20-8699.85$\\\\\n &   &           &     &                & &            &         &  $\\pm0.8$  & $\\pm0.9$  & $\\pm1.0$  & $\\pm1.0$  & $\\pm0.9$ \\\\\n              \\hline\n              \\multicolumn{14}{l}{$^{(a)}$ Radiative broadening is given by the logarithm (base 10) of the FWHM given in $\\rm rad\\,s^{-1}$.} \\\\\n              \\multicolumn{14}{l}{$^{(b)}$ \\citet{Anstee95} notation for the broadening cross-section ($\\sigma$) for collisions by H\\,I at 10\\,$\\rm km\\,s^{-1}$ and its velocity dependence ($\\alpha$).} \\\\\n             \\multicolumn{14}{l}{$^{(c)}$ Stark broadening constant.} \\\\\n               \\multicolumn{14}{l}{$^{(d)}$ \\citet{BWL}, $^{(e)}$ \\citet{2014MNRAS.441.3127R}, $^{(f)}$ \\citet{MRW}, $^{(g)}$ \\citet{FMW}, $^{(h)}$ \\citet{RU}, $^{(i)}$ \\citet{GESB82c}.} \\\\\n         \\end{tabular}\n\\end{table*} \n\n\\subsubsection{Energy levels}\n\nEnergy levels were downloaded from Robert Kurucz's online database, updated in 2013 for FeII \\footnote{\\url{http:\/\/kurucz.harvard.edu\/atoms\/2601}} and 2014 for FeI \\footnote{\\url{http:\/\/kurucz.harvard.edu\/atoms\/2600}}. These data are referenced in the VALD3 \\citep{Ryabchikova15} data base as \\citet[][\"K13\"]{K13} and \\citet[][\"K14\"]{K14}, respectively, and include both observed and theoretically predicted energy levels. The importance of the inclusion of predicted energy levels was demonstrated by \\citet{Mashonkina11b}. There are 2,980 energy levels of FeI below the first ionisation potential ($63,737\\,\\rm cm^{-1}=7.902\\,eV$), approximately two thirds of which have not been observed. For FeII we consider the 116 energy levels below 60,000\\,$\\rm cm^{-1}$, all of them observed, as more highly excited levels  are not relevant in late-type stellar atmospheres (the second ionisation potential of Fe is $130,655\\,\\rm cm^{-1}=16.199\\,eV$). \n\nWe have homogenised the nomenclature of electron configurations and terms to enable energy levels to be merged. Energies with terms given in jj-coupling notation, e.g. \"2+[1+]\" have instead been designated by the leading eigenvector's term in LS-coupling notation, e.g. \"7F\".  This convention is used in the creation of the term diagrams shown in Fig.\\,\\ref{fig:Fe804} and Fig.\\,\\ref{fig:Fe463}.\n\n\\subsubsection{Transition probabilities}\n\nThe \\citet{K13,K14} database contains 533,772 radiative transitions between bound levels of FeI and 1174 between the bound levels we consider for FeII. We have cross-referenced these data with laboratory measurements of transition probabilities carried out since the late 1970's and identified 2080 matches  (0.4\\% of all lines) for FeI and 115 matches for FeII (10\\% of all lines). The references used for FeI are \\citet{BIPS,GESB79b,GESB82c,GESB82d,GESB86,BKK,BK,BWL,2014ApJS..215...23D,2014MNRAS.441.3127R} and for FeII we use the re-normalised compilation by \\citet{Melendez10}. The source with smallest quoted uncertainty was adopted for lines with multiple sources. \n\nFig.\\,\\ref{fig:fvalues} compares theoretical and experimental values of $\\log(gf)$ for both ionisation states. We find that the agreement is typically better for strong transitions; for $\\log(gf)_{\\rm K14}>-2$, theoretical values for FeI show a bias and scatter with respect to experiment of $0.08\\pm0.29$\\,dex, which increases in magnitude and changes sign to $-0.40\\pm0.79$\\,dex at $\\log(gf)_{\\rm K14}<-2$. For weak lines, there appears to be a correlation with the energy of the upper level involved in the transitions, such that the disagreement is very strong for lines with highly excited upper energy levels, while the least excited are in as good agreement with theory as stronger lines. For FeII lines, we find a bias of $-0.11\\pm0.24$\\,dex. The comparison suggests that the use of theoretical data for diagnostic lines should be avoided for precision spectroscopy. However, sensitivity tests that we carried out indicate that Fe NLTE level populations in the Sun are not sensitive to uncertainties in oscillator strengths of this magnitude. All lines selected for abundance analysis in Sect. \\ref{sect:feabund} have laboratory measurements of $\\log(gf)$. \n\n\n\\subsubsection{Photo-ionisation cross-sections}\n\nWe computed total and partial (state-to-state) photoionisation cross-sections for Fe I with the R-matrix method for atomic scattering as implemented in the RMATRX package \\citep{Berrington95}.  These calculations employed close coupling expansion of 157 states of the Fe II target ion from 35 configurations made by atomic orbitals up to principal quantum number $n=6$. The atomic dataset includes cross-sections for 936 LS terms of Fe I with $n\\le 10$ and $l\\le 7$. Details of this calculation will be presented elsewhere (Bautista \\& Lind 2016, in preparation). This calculation is considerably larger and more accurate than our previous computations of atomic data in \\citet{Bautista97}. We use the total, not partial, photoionisation cross-sections in our model atom to limit the number of bound-free transitions.  Each FeI level is thus bound to a single FeII level, as shown in Fig.\\,\\ref{fig:Fe463}.\n\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.67,viewport=2cm 10.5cm 26cm 20.0cm]{Fe804}\n\\caption[]{The complete Fe model atom without fine structure. FeI levels are shown below the dashed line, which indicates the first ionisation potential, and the associated terms are listed at the bottom x-axis. The FeII levels considered in this work are shown above the dashed line and the associated terms are listed at the top axis. Even parity terms are displayed in red and odd parity terms in blue. The left-hand panel shows all radiative bound-bound transitions and the right-hand panel shows all bound-free transitions.} \n\\label{fig:Fe804} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.67,viewport=2cm 10.5cm 26cm 20.0cm]{Fe463}\n\\caption[]{Same as in Fig.\\,\\ref{fig:Fe804}, but for the reduced model atom used for 3D NLTE calculations. Merged levels are indicated with longer horisontal lines.} \n\\label{fig:Fe463} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.60]{clvprof}\n\\caption[]{Normalised observed (bullets) and synthetic (red lines) centre-to limb profiles for two iron lines, where the numbers below each spectrum correspond to the approximate $\\mu$-angle. The synthetic line profiles have been computed in 3D NLTE and the iron abundance has been calibrated for each line to match the disk centre intensity. The calibrated abundance used for synthesis is indicated at the bottom of each panel. Both observed and synthetic spectra have been radial-velocity corrected so that the line centres coincide with the rest wavelength. Spectra for $\\mu\\ge0.4$ have been incrementally offset vertically by $+0.2$. } \n\\label{fig:clvprof} \n\\end{center} \n\\end{figure*}\n\n\\subsubsection{Electron collisions}\nWe adopt the results of \\citet{Zhang95}, who used the R-matrix method to compute collision rates between electrons and 18 low-excitation states of singly ionised Fe. When not available for bound-bound and bound-free electron impact collisions, we follow the semi-empirical recipes given by \\citet{Allen00} for FeI and FeII, which are originally from \\citet{vanRegemorter62} and \\citet{Bely70}. The same formula was used for optically allowed and forbidden transitions, assuming $f=0.005$ for the latter, which gives the two types of transitions similar efficiencies. A comparison between rate coefficients computed by van Regmorter and \\citet{Zhang95} for bound-bound FeII transitions gives a root mean square deviation of $0.6$\\,dex in the temperature interval $3,000-10,000$\\,K. For bound-free transitions, \\citet{Allen00} mentions a probable uncertainty of 0.3\\,dex. We note that more recent collisional data for FeII now exist and should be used for NLTE calculations \\citep{Bautista15}. For the Sun, NLTE effects on FeII lines are insignificant, so the new data would not influence our results. \n \n\\subsubsection{Hydrogen collisions} \nCollision rates for excitation processes, Fe($\\alpha ^{2S+1}L$) + H($1s$) $\\rightarrow$ Fe($\\alpha'^{2S'+1}L'$) + H($1s$), and charge transfer processes, Fe($\\alpha^{2S+1}L$) + H($1s$) $\\rightarrow$ Fe$^+$($\\alpha'^{2S'+1}L'$) + H$^-$, due to low-energy hydrogen atom collisions on neutral iron have been calculated with the asymptotic two-electron method presented by \\citet{Barklem16}. The calculation used here includes 138 states of FeI, and 11 cores of FeII, leading to the consideration of 17 symmetries of the FeH molecule.  These data will be the subject of a future publication (Barklem, in prep.).\n\nFor transitions with no data available, we approximated values using robust fits to the behaviour of the (logarithmic) quantum mechanical rate coefficients with transition energy at a given temperature. Linear fits were used for de-excitation rates and second order polynomials were used for charge exchange rates. The dispersion around the fits are approximately 1.2\\,dex in the temperature interval $3,000-10,000$\\,K. A more elaborate discussion about the appropriate functional forms of such fits is given by Ezzeddine et al. (submitted).\n\n\\subsection{Atom reduction}\n\\label{sect:atomred}\n\nIn its complete form, our Fe model atom contains more than 3,000 fine-structure energy levels, coupled by half a million radiative transitions. To establish the statistical equilibrium using this atom would mean having to solve the radiative transfer equation for at least hundreds of thousands of frequency points, which is not feasible in 3D. The atom must therefore be simplified, while preserving the overall NLTE behaviour.\n\nThe traditional method used to reduce the size of complex model atoms is to merge close energy levels, implicitly assuming that the levels have the same departure coefficients. The degeneracies of the levels that are merged are used as weights in the calculation of the mean energy and the radiative transition probabilities corresponding to the merged level, in such a way that the sum of $gf$ is preserved. We start by following this approach for the collapse of the fine-structure levels, resulting in 762 bound levels of FeI, 41 levels of FeII, and the FeIII ground state. These levels are coupled by 92,567 transitions between bound states of FeI and 226 transitions between bound states of FeII. All FeI levels are coupled to a core FeII state and the photoionisation cross-sections are tabulated over 1,000-2,000 frequency points each. This model will be used as reference model atom and its term diagram is illustrated in Fig.\\,\\ref{fig:Fe804}. The simplification process has so far preserved level configuration, term, and parity for all levels.\n\nWe thereafter proceed to test how much further the atomic level structure can be simplified without causing a significant change in the departure coefficients. We use the $\\rm\\langle3D\\rangle$ structure of the Sun as the default test model in this section, adopting a depth-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$. Above a certain energy limit, FeI energy levels are now merged that share the same multiplicity, parity, and configuration. We gradually decrease this energy limit, while monitoring the difference in equivalent width with respect to the reference model atom, for lines between $200$\\,nm and $2\\rm\\mu m$. The number of levels were thereby reduced from 804 to 463. \n\nIf all radiative transitions were kept, the 463 level atom would still contain approximately 37,000 transitions. However, many transitions do not contribute significantly to make the level populations depart from LTE. To reduce the number of transitions and enable full 3D calculations, we first computed the net radiative imbalance for each transitions in the $\\rm\\langle3D\\rangle$ model of the Sun, assuming LTE populations, i.e., $\\Delta_{ij}=|n_{i}R_{ij}-n_jR_{ji}|$. We then selected a point in the atmosphere ($\\tau_{\\rm500\\,nm}\\approx0.01$), where the NLTE effects are noticeable and relevant for line formation, and removed radiative transitions with a relatively small value of $\\Delta_{ij}$. Thereby, only 3,000 bound-bound transitions and 100 bound-free transitions were kept. We note that the choice of reference depth does not strongly influence which transitions are discarded. Finally, the wavelength grids of the photo-ionisation cross-sections were down-sampled heavily, to a factor 30 fewer points. The final reduced atom contains approximately 17,000 frequency points and preserves $\\rm\\langle3D\\rangle$ equivalent widths for the Sun within 0.01\\,dex, compared to the reference atom. We note that, within these small uncertainties, the smaller atom gives slightly less efficient over-ionisation of FeI than the larger atom, but that further merging of energy levels would have the opposite effect because the collisional coupling between FeI to the FeII reservoir is reduced.      \n\n\\subsection{Spectral synthesis}\n\\label{sect:spec}\nThe restricted NLTE problem, which neglects feedback effects on the atmospheric temperature and density structure, is solved using the 3D radiative transfer code \\textsc{Multi3D}, developed by \\citet{Botnen97} and \\citet{Leenaarts09}. \\citet{Amarsi16a} and Paper III describe a range of improvements recently made to the code, most importantly a new equation-of-state and background opacity package, frequency parallelisation, and improved numerical precision.  We use the same version of the code and same settings here as described in Paper III, except that we use a finer angle quadrature for the radiative transfer solution while the system converges. The Carlson A4 quadrature has 24 angles in total, four azimuthal and six inclined to the normal direction \\citep{Alder63}. After the level populations have converged, the final spectrum is computed at $\\mu=0.2, 0.4, 0.6, 0.8$ and 1.0, in four azimuthal directions. \n\n\\textsc{Multi3D} calculations were performed on three atmospheric snapshots drawn from the most recent 3D radiation-hydrodynamical simulation with the \\textsc{Stagger} code \\citep[e.g.][]{Stein98,Collet11b,Magic13}. A detailed description of the updated simulation run will be given in a future paper (Amarsi et al. in prep.). The snapshots were resized from their original $240\\times240\\times230$ resolution to $60\\times60\\times101$, as described and tested for an earlier Solar simulation by e.g. \\citet{Amarsi17}. The physical sizes of the snapshots are $6\\times6\\times1.5$Mm.\n\nIn addition to LTE and non-LTE line profiles computed with \\textsc{Multi3D}, we computed line profiles from a larger number of 15 snapshots in LTE using \\textsc{Scate} \\citep{Hayek11}. Subtle differences, of the order of $2-3\\%$, were noticed in the centre-to-limb behaviour of equivalent widths between the two codes, with the latter more closely resembling observations. We therefore computed our final NLTE profiles by multiplying the NLTE\/LTE profile ratio found by \\textsc{Multi3D} with the LTE profiles computed by \\textsc{Scate}. The average effective temperature of the 15 snapshots is $5776\\pm16$\\,K, close enough for our purposes to the nominal $T_{\\rm eff}=5772$\\,K \\citep{Prsa16}.\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.28]{1panel6151}\n\\caption[]{The coloured image and bar on the right-hand side represent the NLTE\/LTE equivalent width ratio of FeI 6151\\AA\\ at disk centre for a single snapshot from the solar convection simulation. In the up-flowing granules, over-ionisation causes the line to weaken in NLTE, while the inter-granular lanes display the opposite effect. The y-axis on the left-hand side indicates the spatial scale.} \n\\label{fig:ratio} \n\\end{center} \n\\end{figure}\n\n\\section{Results and discussion}\n\\label{sect:results}\n\nIt is well-known that level populations of Fe do not strongly depart from LTE in the line-forming regions of the Sun and NLTE effects on line strengths are therefore small \\citep[e.g.][]{Mashonkina11a,Bergemann12}.  The $\\rm\\langle3D\\rangle$ solar model predicts significant over-ionisation of FeI to be important only at very optically thin layers ($\\log(\\tau_{500\\rm nm})<-3.5$), while over-recombination barely dominates in deeper layers ($-2<\\log(\\tau_{500\\rm nm})<-3$), and even deeper layers are fully thermalised. Line strengths are typically affected by less than 0.01\\,dex. In full 3D, the NLTE effects vary with the convection pattern and all but the highest excited levels experience under-population in the up-flowing granules and over-population in the inter-granular lanes (Fig.\\,\\ref{fig:ratio}). This variation is expected given the much steeper temperature gradients of the granules and the behaviour is qualitatively similar to that found by \\citet{Shchukina01}, although they predict stronger over-ionisation overall. This difference is likely due to the model atoms; our atom contains many more highly excited levels and collisions with neutral hydrogen, which strengthen the collisional coupling between FeI and the FeII reservoir and reduces NLTE effects. The surface variation can also be compared to the NLTE effects of Li\\,I, Na\\,I, Mg\\,I and Ca\\,I in metal-poor stars \\citep{Asplund03,Lind13,Nordlander16}. The net effect from our 3D NLTE modelling is more over-ionisation of FeI compared to the $\\rm\\langle3D\\rangle$ model and low-excitation lines in particular are substantially weakened. In Sect.\\,\\ref{sect:feabund}, we describe how these effects propagate into iron abundance corrections.    \n\nFor three FeI lines, we can compare our predicted NLTE effects with those of \\citet{Holzreuter13}. Their Fig.\\,8 shows histograms of the equivalent-width ratios between LTE and NLTE at each pixel in the $xy$-plane for FeI 5250\\AA , 6301\\AA , and 6302\\AA .  For the bluer line, which has low excitation potential, we find a mean ratio of $+4$\\%; significantly less than their $+15$\\%. For the redder lines, we find $-1$\\%, compared to their $+1$\\%. Again, differences in model atom structure and adopted collisional cross-sections are most likely responsible for their stronger over-ionisation. \n\n\\subsection{Centre-to-limb variation}\n\\label{sect:clv}\nAfter performing an assessment of blends, we selected eleven iron lines, including one FeII line, within the SST wavelength ranges (See Table \\ref{tab:lines}). The ten FeI lines span a wide range in wavelength and strength, but unfortunately a narrow range in lower level excitation potential. As mentioned above, low-excitation lines are most sensitive to NLTE effects, but the only observed line, 5371\\AA , connected to a level below $2\\,\\rm eV$ in our wavelength regions is too blended to have diagnostic value and we therefore excluded it. \n\nThe centre-to-limb behaviour is depicted in Fig.\\,\\ref{fig:CLVew}. The observed data points correspond to average equivalent widths measured  at each $\\mu$-angle and the vertical error bars to the standard deviation of the individual pointings added to an estimated 0.5\\% error due to continuum placement. Equivalent widths were measured by direct integration within wavelength ranges that were considered blend-free (see Table \\ref{tab:lines}), after applying a radial velocity correction that aligns the deepest point of the line profile with the rest wavelength. The curves correspond to the predicted equivalent widths at a given abundance for each model and line, optimised to match disk-centre line strengths. The model spectra were similarly corrected to rest wavelength and integrated over the same wavelength interval as the observations. We chose this approach to enable a comparison between the models that is as fair as possible, because the 3D velocity field gives rise to a differential radial velocity effect with $\\mu$ that is not captured in 1D or $\\rm\\langle3D\\rangle$.\n\nFull 3D modelling matches the observed centre-to-limb behaviour well; the equivalent widths are reproduced to within $\\sim$5\\% in both LTE and NLTE.  Comparing the two, the latter performs better for strong lines, 5367\\AA\\ and 5383\\AA , and for the only line that becomes weaker in NLTE, 6151\\AA , which has the lowest excitation potential of our lines. LTE is slightly better for 5373\\AA\\ and 5389\\AA\\ , but the differences are small and restricted to $\\mu=0.2$. There is a general tendency for the 3D equivalent widths of weak lines, $W_\\lambda<100\\rm\\,m\\AA$, to be over-predicted by a few percent at $\\mu=0.2$.  \n\nWe have investigated if a better match to the limb observations could be achieved by modifying the model atom. A single 3D snapshot was run with model atoms for which all hydrogen collision and electron collision rates, respectively, were reduced by an order of magnitude. The results for the atom with modified hydrogen collisions is labeled $\\rm H\\times0.1$ in Fig.\\,\\ref{fig:CLVew} and the atom with modified electron collisions is labeled $\\rm e\\times0.1$.  We find that reduced hydrogen collisions systematically strengthen the limb equivalent widths compared to the disk centre, such that the discrepancy with the observations increases for most lines. The effect on the level populations is such that the departures from LTE are simply shifted to deeper layers. Reducing the electron collisions also makes NLTE effects set in at deeper layers, but it also gradually enhances the over-ionisation of FeI with decreasing atmospheric depth. This has a small differential effect on the centre-to-limb variation that improves the agreement with observations in most cases. \n\nThe change in line strength as a function of viewing angle is not well predicted by the $\\rm\\langle3D\\rangle$ model, which gives systematically too small equivalent widths at the limb compared to the line centre. NLTE line formation alleviates the problem slightly for FeI lines, but the line strength at $\\mu=0.2$ is still $5-20$\\% too small. The different behaviour to full 3D modelling can be largely attributed to the treatment of velocity fields; Fig.\\,\\ref{fig:CLVew2} shows the results of 3D LTE modelling with the velocity field at all points and in all directions set to zero, but with a constant microturbulence of $1\\rm\\,km\\,s^{-1}$. Evidently, this method reproduces the $\\rm\\langle3D\\rangle$ centre-to-limb behaviour very closely, in particular for the FeII line and the high-excitation FeI lines. Fe\\,I 6151\\AA\\ shows a slightly larger difference, which is probably caused by its lower excitation potential and thus greater sensitivity to temperature inhomogeneities. Fig.\\,\\ref{fig:CLVew2} also shows the results of using a 1D \\textsc{MARCS} model atmosphere \\citep{Gustafsson08} with $1\\rm\\,km\\,s^{-1}.$ microturbulence, which even more strongly underestimates the the line strengths at the limb, in agreement with the Fe line analysis of \\citet{Pereira09b}. The difference with respect to $\\rm\\langle3D\\rangle$ may be attributed to the slightly steeper temperature gradient around continuum optical depth unity.\n\nThis failure of 1D models is well-known and was reported already by \\citet{Holweger78}, who demonstrated that a $\\mu$-dependent microturbulence may solve the problem. Their Fe line analysis found empirically that a value of $1.6\\rm\\,km\\,s^{-1}$ is suitable at $\\mu=0.3$, compared to $1.0\\rm\\,km\\,s^{-1}$ at the disk centre, thus strengthening lines at the limb compared to centre. The same qualitative behaviour of 1D models has also been demonstrated for the centre-to-limb behaviour of the O\\,I 777\\,nm triplet \\citep{Steffen15}. We refrain from deriving an empirical $\\mu$-dependent microturbulence for 1D and $\\rm\\langle3D\\rangle$ modelling to match our observations, but emphasize that models can now predict the 3D velocity field and thus the line broadening without invoking free parameters. We note that strengthening of lines toward the limb can occur also in 1D models as a consequence of the change in temperature gradient, without considering the velocity field, as shown e.g. for very weak ($<15\\rm\\,m\\AA$) O\\,I, Sc\\,II, and Fe\\,I lines by \\citep{Pereira09b}.\n\n\\citet{Mashonkina13} modelled the centre-to-limb behaviour of two FeI lines, 6151\\AA\\ and 7780\\AA , using the SST observations of \\citet{Pereira09b}. They report 1D LTE modelling based on \\textsc{MAFAGS-OS} model atmosphere \\citep{Grupp09}, 3D LTE modelling based on a \\textsc{CO$^5$BOLDT} model atmosphere \\citep{Freytag12}, and $\\rm\\langle3D\\rangle$ LTE and NLTE modelling. For the bluer line, also studied in this paper, their 1D and $\\rm\\langle3D\\rangle$ LTE results are in good agreement with ours, but their 3D LTE modelling predicts more line strengthening toward the limb, implying that the 3D velocity field is characteristically different from our \\textsc{Stagger} model. For the redder line, not studied here, they find $\\rm\\langle3D\\rangle$ NLTE to well reproduce the centre-to-limb behaviour.\n\n{The importance of velocity fields aside, little can be found in the literature to explain the model centre-to-limb behaviour of different lines from basic principles. In general, we find that the $\\rm\\langle3D\\rangle$ model predicts line strengthening toward the limb for blue lines  ($<4000\\AA$) and line weakening for red lines. Strong lines tend to be more weakened than weak lines at a given wavelength, as can be seen in Fig.\\,\\ref{fig:CLVew}. We find that this behaviour can be partly explained by equation 17.183 in \\citet{Hubeny14}:\n\n\\begin{equation}\n\tr_\\nu(\\mu)\\equiv I_\\nu(0,\\mu)\/I_c(0,\\mu)=[a_\\nu+b_\\nu\\mu\/(1+\\beta_\\nu)]\/(a_\\nu+b_\\nu\\mu)\n\\end{equation}\n\nTo derive this expression, the authors assume a line formed in true absorption and a linear dependence of the Planck function with continuum optical depth at a given frequency, such that $B_\\nu=a_\\nu+b_\\nu\\tau_c$, where $a_\\nu$ and $b_\\nu$ are positive constants. {Scattering and velocity fields are neglected. $I_\\nu(0,\\mu)$ is the emergent intensity at a given $\\mu$-angle, $I_c(0,\\mu)$ is the corresponding continuum intensity, and $\\beta_\\nu$ is the ratio between line and continuous opacity. Since $1+\\beta_\\nu>1$, the residual intensity at the limb is always higher than at disk centre for a given wavelength. Lines are therefore always predicted to weaken with decreasing $\\mu$, which we have seen is true at least for red lines according to $\\rm\\langle3D\\rangle$ modelling. It may further explain why strong lines typically decrease more in line strength than weak lines, because the inverse dependence on $\\beta_\\nu$ has higher influence on the residual intensity at higher $\\mu$. When $\\beta_\\nu\\gg1$, Eq. 1 approaches $a_\\nu\/(a_\\nu+b_\\nu\\mu)$, which implies that the behaviour for strong lines at a given frequency is similar. This is true for the two strongest lines in our sample. Further, we estimated values for the coefficients $a_\\nu$ and $b_\\nu$ in the region around continuum optical depth unity for our lines and found that they change in such a way that it can explain why redder lines are more weakened than bluer (see Fig.\\,\\ref{fig:CLVew} and \\ref{fig:CLVew2}) . However, the dependence is weaker than what the detailed modelling predicts. The validity of Eq.\\,1 thus appears limited by the assumptions made.  \n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.8]{CLVew}\n\\caption[]{Centre-to-limb variation of solar iron lines. The black bullets are observed equivalent widths and the lines represent predictions in LTE and NLTE for different model atmospheres and atomic data. A depth- and $\\mu$-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$ was adopted for the $\\rm\\langle3D\\rangle$ models. In the $\\rm H\\times0.1$ and $\\rm e\\times0.1$ models, Hydrogen and electron collisional rates were reduced by a factor ten, respectively.} \n\\label{fig:CLVew} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.8]{CLVew2}\n\\caption[]{The black bullets and red dashed lines are the same as in Fig.\\,\\ref{fig:CLVew}. The red solid lines represent 3D LTE modelling without velocity fields. For all models, we assume a depth- and $\\mu$-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$. }\n\\label{fig:CLVew2} \n\\end{center} \n\\end{figure*}\n\n\\subsection{Iron abundance}\n\\label{sect:feabund}\n\\citet{Scott15} revised the solar iron abundance of \\citet{Asplund09} using disk-centre intensities of 31 FeI and FeII lines, carefully selected based on blending properties, line strength, and atomic data.  They employed an earlier version of a 3D hydrodynamical \\textsc{Stagger} simulation of the solar photosphere and the same Fe model atom as in Papers I and II in this series.  Abundances were first computed in 3D LTE and then corrected using NLTE calculations based on a $\\rm\\langle3D\\rangle$ model. \\citeauthor{Scott15} recommended a weighted mean abundance $\\log(\\epsilon_{\\rm Fe})=7.47\\pm0.04\\rm\\,dex$. They find that the excitation balance of FeI is well established in 3D LTE, whilst the NLTE abundances show a slight negative trend with excitation potential. FeI and FeII lines give a difference in mean abundance of 0.07\\,dex in LTE, which decreases to 0.06\\,dex after NLTE corrections have been applied. \n\nIn this study, we re-determined iron abundances for the lines selected by \\citeauthor{Scott15}, but using consistent 3D NLTE modelling. As described in the beginning of Sect.\\,\\ref{sect:results}, full 3D calculations predict a higher degree of over-ionisation than $\\rm\\langle3D\\rangle$ calculations. Our new analysis technique and new model atomic data for Fe result in more positive abundance corrections for low-excitation FeI lines; between $+0.03$ and $+0.06\\rm\\,dex$ for $E_{\\rm low}<1\\rm\\,eV$), while high-excitation lines ($E_{\\rm low}>4\\rm\\,eV$) are at most affected by $-0.01$\\,dex. This can be compared to $+0.11$\\,dex and $+0.06$\\,dex predicted for low and high-excitation lines, respectively, by \\citet{Shchukina01}. Our FeI line abundances move slightly further away from fulfilling excitation balance, while the offset in ionisation balance is reduced to 0.04\\,dex. The weighted mean abundance of all lines becomes slightly larger; $7.48\\pm0.04$\\,dex.\n\nWe repeated the model atom modifications described in Sect.\\,\\ref{sect:clv}, in order to evaluate if better agreement between different iron lines can be achieved. The $\\rm H\\times0.1$ model with altered hydrogen collisions improves neither excitation nor ionisation balance, while the $\\rm e\\times0.1$ model with altered electron collisions reduces the ionisation imbalance to 0.01\\,dex, but at the expense of further strengthening the excitation imbalance. Turning to other potential sources of error, we note our use of electron densities computed in LTE, although important electron donors (including hydrogen) have been shown to have significant NLTE effects. We also remind the reader that the atom reduction itself may have a small impact (see Sect.\\,\\ref{sect:atomred}).   \n\nFinally, \\citet{Scott15} discussed the influence on FeI line abundances by magnetic fields, referencing the work of \\citet{Fabbian12}, and concluded that an ionisation imbalance of order 0.02\\,dex may be amended by using realistic magneto-hydrodynamic simulations with an average field strength of 100\\,G \\citep{TrujilloBueno04}. However, the simulations by \\citet{Moore15} showed that the magnetic field must be concentrated and coherent to have an impact; a small-scale, randomly oriented field of 80\\,G would not affect the iron abundance determination significantly. \\citet{Shchukina15} concluded, based on the magneto-convection simulation by \\citet{Rempel12}, that a small-scale dynamo with no net magnetic flux would have a typical influence on FeI line abundances of the order $+0.014$\\,dex. \n\n\\section{Conclusions}\n\\label{sect:conc}\n\nWe have demonstrated that full 3D, NLTE modelling of iron line formation of the Sun, using a comprehensive model atom with 463 levels, is now feasible and can successfully reproduce observed data without invoking free parameters  \\citep[see also Paper III and][]{Nordlander16}. In particular we conclude here:\n\n\\begin{itemize}\n\n\\item 3D NLTE effects on low-excitation FeI lines ($<1$\\,eV) are stronger than predicted by $\\rm\\langle3D\\rangle$ modelling, resulting in 0.03-0.06\\,dex higher abundances for these lines.\n\n\\item When normalised to disk-centre line strength, full 3D NLTE modelling typically over-predicts limb (here $\\mu=0.2$) line strengths by approximately 5\\,\\%. 1D and $\\rm\\langle3D\\rangle$ modelling in LTE and NLTE perform significantly worse, assuming a constant microturbulence of $1\\rm\\,km\\,s^{-1}$, independent of depth and viewing angle. We stress the importance of proper treatment of the 3D velocity field for centre-to-limb modelling. \n\n\\item The iron abundance of the Sun is found to be $\\log(\\epsilon_{\\rm Fe})=7.48\\pm0.04$\\,dex, using consistent 3D NLTE modelling of the lines selected by \\citet{Scott15}.\n\n\\item The ionisation imbalance between FeI and FeII line abundances in the Sun is reduced to 0.04\\,dex compared to 0.06\\,dex found by \\citet{Scott15}. FeI line abundances show a negative slope with respect to excitation potential, similarly to metal-poor standard stars (see Paper III).\n\n\\item Rates of collisional excitation and ionisation of FeI by electrons still rely on simple semi-empirical recipes. Our tests show that less efficient electron collisions than employed in this work can improve agreement with solar observations in certain respects. This highlights the urgent need of improved data for such transitions, e.g.\\ using the R-matrix method. \n\n\\item High-quality solar observations at different viewing angles pose excellent challenges for spectral line formation models, testing the accuracy of atomic data as well as physical assumptions. Low-excitation FeI lines are of particular diagnostic importance and more data should be obtained.    \n\n\\end{itemize}\n\n\n\\section*{Acknowledgments} \nKL acknowledges funds from the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the Federal Ministry of Education and Research as well as funds from the Swedish Research Council (Grant nr. 2015-00415 3) and Marie Sk\\l odowska Curie Actions (Cofund Project INCA 600398). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at UPPMAX under project p2013234. AMA and MA are supported by the Australian Research Council (grant FL110100012). PSB acknowledges support from the Royal Swedish Academy of Sciences, the Wenner-Gren Foundation, G\\\"oran Gustafssons Stiftelse and the Swedish Research Council. For much of this work PSB was a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. PSB is presently partially supported by the project grant The New Milky Way from the Knut and Alice Wallenberg Foundation. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). TMDP was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7\/2007-2013) \/ ERC Grant agreement No. 291058. The Swedish 1-m Solar Telescope was at the time of our observations operated on the island of La Palma by the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof\\'{i}sica de Canarias. Finally, we thank the Max Planck Institute for Astrophysics in Garching for our SST observing time. \n\n\\bibliographystyle{mn2emod} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe discovery of quasars beyond $z\\sim7$ (eg: \\cite{banados2018800,mortlock2011luminous}) poses a crucial question: which cosmic era marks the birth of radio-loud \\ac{AGN}? \nRadio loud \\ac{AGN}s are amongst the most radio-luminous sources at all cosmic epochs. Their large radio luminosity is attributed to a radio-jet launched by an accreting \\ac{SMBH} at their centre. So far, radio-\\ac{AGN}s are only known upto $z\\sim6$, and the most distant radio quasar is at $z=6.82$ \\citep{banados2021discovery}, while the  highest redshift radio galaxy lies at $z=5.72$ \\citep{saxena2018discovery} and the highest-redshift blazar is at $z=6.1$ \\citep{belladitta2020first}. Of $\\sim$200 quasars discovered beyond $z\\sim6$, only five are found to be radio sources.\n\n\\ac{HzRG}s have been targeted either by looking for ultra steep ($\\alpha < -1.3$) radio spectra \\citep[USS, ][]{de2000sample}, or by selecting sources with a very faint K-band (2.2\\,$\\mu$m) counterpart \\citep{jarvis2009discovery}. The former technique is the most widely used and is based on the observed steepening of the radio spectrum with redshift. This method has been used to discover almost all known \\ac{HzRG}s, including the most distant known source at $z = 5.72$ \\citep{saxena2018discovery}. However, some studies \\citep{yamashita2020wide,jarvis2009discovery,miley2008distant} have reported the discovery of HzRGs with non-steep spectral index at $z \\ge 4$, showing that the USS selection method does not give a complete view of high-redshift radio-\\ac{AGN}s. \n\n\nSimulations of forthcoming radio surveys estimated the source count of radio emitters as a function of redshift \\citep{bonaldi2019tiered,wilman2008semi} and it predicts hundreds of thousands of radio sources beyond $z \\sim 4$. Although radio emitters such as radio-loud \\ac{AGN}s, Starbursts, \\ac{SFG}s and radio-quiet \\ac{AGN}s contribute to this total radio-source count, simulations demonstrate that $z\\ge4$ sky is dominated by radio-AGNs (see \\cite[Figure~2]{raccanelli2012cosmological}).\nCurrently, $\\sim$112 published radio-\\acp{AGN} are known at redshift $z \\gtrsim 4$ (listed in Appendix, Table~\\ref{tab:known_radio_AGN}). This number thus indicates that the known radio source population at $z\\gtrsim4$ represents a small fraction of the total radio source population.\n\nIn some cases, it is unclear whether a  detected high-redshift radio source is a radio galaxy, blazar, quasar, etc., and so we adopt the neutral term ``\\ac{HzRS}'' to describe any radio source detected at high redshift ($z\\ge4$) .\n\nThe mismatch between models and data indicates that known \\acp{HzRS} are only the tip of the iceberg. The dearth of radio sources at high redshift can be attributed to the following factors: \n(i) many \\acp{HzRS} are probably in existing radio catalogues but their redshifts have not been measured  due to \ntheir faintness at optical\/IR wavelengths, \nand (ii) previous radio surveys were not sensitive enough to detect faint \\acp{HzRS}\n\n\n Since the ultimate goal of this series of papers is to establish the \\ac{HzRS} count and thus %\ntest the simulations\n\\citep{bonaldi2019tiered,wilman2008semi}, we expect that some missing \\acp{HzRS} are already in the literature, but are not classified as high-redshift radio sources.  \n We demonstrate this by visually cross-matching \\ac{SDSS} spectroscopy \\citep[DR12,][]{sdss} with the \\ac{FIRST} \\citep{becker95} and \\ac{NVSS} \\citep{condon98} catalogues. This search resulted in a further 33 sources at $z \\gtrsim 4$, listed in Appendix, Table \\ref{tab:new_radio_AGN}. In each case the \\ac{SDSS} spectrum has been checked for supporting evidence of the redshift, such as a Lyman break or other spectral features. We note that a further list of candidates is available in the MILLIQUAS \\citep{milliquas} catalogue\\footnote{\\url{https:\/\/heasarc.gsfc.nasa.gov\/W3Browse\/all\/milliquas.html}}, but to the best of our knowledge the spectroscopy has not been checked and so that list may include some spurious candidates.\n\n\\ac{EMU} is one of the deepest and the largest forthcoming radio continuum surveys \\citep{norris2011emu}, to be delivered by the \\ac{ASKAP} telescope \\citep{johnston2007science}. The \\ac{EMU} project started with a series of Early Science observations, followed by the \\ac{EMU} Pilot Survey \\citep[PS;][]{norris21}. In this paper, we use the \\ac{EMU} Early Science Observations of the GAMA23 field (hereafter referred to as ``G23''), which is one of the Galaxy And Mass Assembly (GAMA) survey fields \\citep{driver2008galaxy}. \n\nMotivated by the challenge of finding missing \\acp{HzRS}, we make use of a search technique different from conventional radio based techniques, the Lyman Dropout technique  ( a.k.a. Lyman Break Galaxy technique), to identify potential \\acp{HzRS} at $ z \\gtrsim 4-7$ in the G23 field. The Lyman dropout technique has been a popular technique in optical astronomy over the past two decades for discovering high-redshift galaxies up to  $z\\sim11$. However, only one radio galaxy at $z=4.72$ has been identified using the Lyman dropout technique to date \\citep{yamashita2020wide}.\nTherefore, the primary goal of this study is to test the efficiency of Lyman dropout technique in finding \\acp{HzRS}. \nA second goal is to determine the properties of our sample of \\acp{HzRS}, \n a detailed study of which will be discussed in a future paper. \n\nThe Lyman dropout technique looks for the redshifted spectral signature of the \\emph{Lyman limit} at 91.2\\,nm (Far-UV regime). This is the longest wavelength of light  that can ionise a ground-state hydrogen atom. Light at wavelengths shorter than 91.2\\,nm (ie. at higher energies) will be absorbed by sufficiently optically-thick atomic hydrogen present in the galaxy or its circumgalactic medium. This missing radiation creates a break in the observed spectrum. For high-$z$ galaxies, the Lyman break gets redshifted into the optical region, and can be identified using images taken in multiple filters.\n\nThe structure of this paper is as follows. In Section~\\ref{sec:data} we describe how we select \\acp{HzRS} in GAMA23 field from the \\ac{ASKAP} 887.5\\,MHz radio catalogue using 8-band \\textit{ugriZYJK\\textsubscript{s}} KiDS\/VIKING photometry. In Section~\\ref{sec:result}, we present our sample of \\ac{HzRS} candidates selected at $z \\sim 4$, 5, 6, and 7. In Section~\\ref{sec:discuss} we present the analysis of radio and IR properties of our sample. Finally, we summarise our results in Section~\\ref{sec:concl}.\n\nThis study adopts a $\\Lambda$CDM cosmology with $\\Omega\\textsubscript{m} = 0.3$, $\\Omega\\textsubscript{$\\Lambda$} = 0.7$ and $H_0 = 70~$km~s$^{-1}$Mpc$^{-1}$. \n\n\n\n\\section{Data and Methods}\n\\label{sec:data}\n\nTo find \\acp{HzRS}, we cross-match the G23 radio observations with the Kilo Degree Survey optical catalogue \\citep[KiDS;][]{kuijken2019fourth} and the VIKING DR5 \\& CATWISE2020 \\citep{marocco2021catwise2020} infrared catalogues. This results in a sample of G23 radio sources with optical and infrared photometry. We then apply the redshift specific Lyman dropout colour cuts \\citep{ono2018great,venemans2013discovery} to select the radio source candidates at $z \\gtrsim 4$ in the G23 field. This paper is the first in a series describing our search for \\acp{HzRS} using the Lyman dropout technique as part of the \\ac{EMU} survey.\n\nWe use the 887.5\\,MHz radio continuum data of the G23 field, produced by \\ac{ASKAP} as part of the \\ac{EMU} Early Science program in early 2019. \\ac{ASKAP} consists of 36 antennas, each of which is equipped with a Phased Array Feed (PAF). It operates in a frequency range from 700 to 1800\\,MHz. \\ac{ASKAP} data products have been created using the ASKAPsoft pipeline, aided by Selavy software in source extraction. This study examined the following \\ac{ASKAP} catalogues from project AS034: (i) selavy-image.i.SB8132.cont.taylor.0.restored.components and (ii) selavy-image.i.SB8137.cont.taylor.0.restored.components, retrieved from the CSIRO Data Access portal\n\\footnote{\\url{https:\/\/data.csiro.au\/domain\/casdaObservation}}.  The observational parameters of G23-ASKAP data are given in Table~\\ref{tab:survey_info}.\n\n A total of 38\\,080 radio sources are present in these 2 catalogues, of which 2107 are complex or multi-component (number of components $\\ge 2$). In this paper, we focus on simple (or single component) radio sources only, which are fitted by a single Gaussian. \n\n  For the selection of $z\\ge4-6$ radio sources, we used optical data from the complementary Kilo Degree Survey \\citep[KiDS,][]{kuijken2019fourth}, in particular we exploited the KiDS DR4.1 multiband source catalogue, featuring Gaussian Aperture and PSF photometry ( GAaP ; see \\cite{kuijken2015gravitational} for details) measurements of KiDS-$ugri$ and VIKING-ZYJHK bands for $r$-band detected sources.\n\nTo select $z\\sim7$ radio sources, we utilized VIKING photometry in the DR5 catalog obtained from the VISTA archive. \\footnote{\\url{http:\/\/horus.roe.ac.uk\/vsa\/index.html}} \nWe converted the Vega magnitudes in the VIKING DR5 catalog to AB magnitudes using the Cambridge Astronomical Survey Unit (CASU) recommendations \\footnote{\\url{http:\/\/casu.ast.cam.ac.uk\/surveys-projects\/vista\/technical\/filter-set}}.\n \n\nThe mid-IR (MIR) data used in this paper comes from the Wide-field Infrared Survey Explorer \\cite[WISE,][]{wright2010wide}, which is an all-sky survey centred at 3.4, 4.6, 12, and 22\\,$\\mu$m (referred to as bands W1, W2, W3 and W4), with an angular resolution of 6.1, 6.4, 6.5, and 12.0~arcsec respectively, and typical 5$\\sigma$ sensitivity levels of 0.08, 0.11, 1, and 6~mJy\/beam. Here, we use data from the CATWISE2020 \\citep{marocco2021catwise2020} catalogue.\n\nFar-IR (FIR) observations in the G23 field come from the {\\it Herschel} space observatory.  {\\it Herschel} carried out observations using two photometric instruments on board, (i) Photodetecting Array Camera and Spectrometer (PACS, \\cite{poglitsch2010photodetector} ) and (ii) Spectral and Photometric Imaging Receiver (SPIRE, \\cite{griffin2010herschel} ). PACS observations centred at 70\\,$\\mu$m, 100\\,$\\mu$m, and 160\\,$\\mu$m mainly trace the rest-frame mid-IR emission of the high-$z$ ($z>2$) \\ac{AGN}. SPIRE observed simultaneously in three wavebands centred at 250\\,$\\mu$m, 350\\,$\\mu$m, and 500\\,$\\mu$m, picking up the starburst emission in high-$z$ \\ac{AGN}s \\citep{hatziminaoglou2010hermes}. {\\it Herschel} ceased operation on 29$^{\\rm th}$~April~2013 when the telescope ran out of liquid helium, which is essential for cooling the instruments. This study utilized the SPIRE \\citep{schulz2017spire} and PACS point source catalogues \\citep{marton2017herschel} available in \\ac{IRSA}\\footnote{\\url{https:\/\/irsa.ipac.caltech.edu\/}}.\n\n\\begin{table}\n    \\centering\n      \\caption{ Summary of G23-ASKAP survey.}\n    \\begin{tabular}{l|c}\n      Parameters   & G23-ASKAP \\\\\n         \\hline\n       Frequency  (MHz)         & 887.5     \\\\\n       Bandwidth  (MHz)         & 288       \\\\\n       Synth. beam size ($\\,arcsec$) &  10         \\\\\n       RMS ($\\mu$Jy\/beam)   &    38           \\\\\n       Survey area (deg\\textsuperscript{2})  & 50      \\\\\n       Astrometric accuracy ($\\,arcsec$)&      $\\sim$ 1    \\\\\n      \\hline\n    \\end{tabular}\n    \\label{tab:survey_info}\n\\end{table}\n\n\\subsection{Finding optical and infrared counterparts}\n\\label{sec:optical_crossmatch}\n\nTo find the optical counterparts of radio sources, we use a simple nearest-neighbour technique. We need to choose a search radius that maximises the number of cross-matches while minimising the number of false identifications (hereafter called false-IDs). We achieve this by cross-correlating the \\ac{ASKAP} radio catalogue with the KiDS DR4.1 multiband optical (\\textit{ugri}) $+$ NIR (ZYJHK\\textsubscript{s}) photometry at a range of search radii, measuring the number of cross-matches at each radius. \nWe then estimate the false-ID rate by shifting the radio position by 1~arcmin (so that all matches are spurious) and repeating the cross-match at the same set of radii. The false-ID rate is calculated by dividing the number of shifted cross-matches by the number of unshifted cross-matches. The result is shown in Table~\\ref{tab:false_id}. Based on this, we have chosen 2~arcsec as the optimal search radius for this study, corresponding to a false-ID rate of 14.64\\% and a total cross-match rate of 63.9\\%. We reject sources that had multiple matches within 2~arcsec. This reduces the final number of radio-optical cross-matches to 17\\,447.\n\nWe followed the same procedure to select infrared counterparts to our radio sources. We cross-matched the optical (KiDS) positions of our sample with the CATWISE2020 catalog at a search radius of 2$\\,arcsec$. This gives a false id rate of 11.9\\%.\n\n\\begin{table}\n \\centering\n \\caption{False-ID rate estimated for \\ac{ASKAP}-KiDS cross-match as a function of separation radius, using the single-component source list.}\n \\label{tab:false_id}\n \\begin{tabular}{c c c c}\n \\hline\n  Match Radius & No. of Matches & No. of Matches & False-ID Rate \\\\\n  (arcsec) & (unshifted) & (1~arcmin offset) & (\\%) \\\\\\hline\n  1 & 14\\,067 & 833 & 5.92 \\\\\n  2 & 23\\,002 & 3\\,368 & 14.64 \\\\\n  3 & 29\\,350 & 7\\,592 & 25.87 \\\\\n  4 & 36\\,417 & 13\\,410 & 36.82\\\\\n  5 & 44\\,884 & 20\\,821 & 46.39 \\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\n\\subsection{Selection of radio sources at $z\\gtrsim$ 4 -- 6}\n\n The Lyman dropout technique relies on finding the wavelength or passband at which the Lyman break is detected, which in turn tells us the redshift  of the source, given that rest-frame wavelength of Lyman limit is 91.2\\,nm. For example, the Lyman break of a galaxy at $z=4$ will be observed at wavelength 4560\\,\\AA\\,  and hence can be imaged in g-band. Similarly, for higher redshift objects, the Lyman break moves into the $r$ or $i$ or $Z$ bands.\n\\subsubsection{Applying  \\textit{g, r, \\& i}-band  dropout technique}\n\\label{sec:initial_sample}\n\n\n\\begin{table}\n\\caption{KiDS \\& VIKING filters, their central wavelength, and their mean 5$\\sigma$ limiting magnitude. }\n \\centering\n \\begin{tabular}{c c c}\n \\hline\n  Filters & $\\lambda$ & 5$\\sigma$ Mag. Lim. \\\\\n    & (\\AA)    & (AB) \\\\\n    \\hline\n  $u$ & 3\\,550  & 24.23 \\\\\n  $g$ & 4\\,775 & 25.12\\\\\n  $r$ & 6\\,230  & 25.02 \\\\\n  $i$ & 7\\,630  & 23.68 \\\\\n  Z & 8\\,770  &23.1 \\\\\n  Y & 10\\,200 & 22.3 \\\\\n  J & 12\\,520  & 22.1 \\\\\n  H & 16\\,450  & 21.5\\\\\nK\\textsubscript{s} & 21\\,470  & 21.2 \\\\\n\\hline\n \\end{tabular}\n \\label{tab:kiDS_filters}\n\\end{table}\n\n\n\\begin{table*}\n \\begin{threeparttable}\n \\caption{\n The criteria used to identify Lyman dropouts at $z\\sim$ 4, 5, 6, adopted from \\protect \\citet{ono2018great} and at $z\\sim$ 7, taken from \\protect \\cite{venemans2013discovery}. The last row is based on our COSMOS tests, to remove low-z interlopers in $z\\sim$ 4, 5, 6 sample, as described in subsection \\protect \\ref{sec:interloper1}.} \n \n \n \\begin{tabular}{p{2.9cm} | p{2.9cm} | p{2.9cm} | p{2.9cm} |p{3.9cm}}\n \\hline\n  \\multirow{2}{*}{$z\\sim4$ (\\textit{g} dropouts)} & \\multicolumn{2}{c}{$z\\sim5$ (\\textit{r} dropouts)} & \\multirow{2}{*}{$z\\sim6$ (\\textit{i} dropouts)} &  \\multirow{2}{*}{$z\\sim7$ (Z dropouts)} \\\\\n  \\cline{2-3}\n  &  criteria I \\tnote{a} &  criteria II \\tnote{b} & & \\\\\n  \\hline\\hline\n   S\/N(i) $>$5 & S\/N(z) $>$5 & S\/N(z) $>$5 & S\/N(z) $>$5  &  S\/N(Y) $>$7\\\\\n   & S\/N(g) $<$2  & S\/N(g) $<$2 &S\/N(g) $<$2 ; S\/N(r) $<$2 & Z-Y $\\ge$ 1.1  \\\\\n   \\textit{g-r $>$ 1.0} &\\textit{r-i $>$ 1.2}&\\textit{r-i $>$ 1.0} &\\textit{i-z $>$ 1.5} & -\u2212 0.5 $<$ Y \u2212 J $\\le$ 0.5 \\\\\n   \\textit{r-i $<$ 1.0} &\\textit{i-z $<$ 0.7} &\\textit{i-z $<$ 0.5} & \\textit{z-Y $<$ 0.5} & Z \u2212 Y $>$ Y \u2212 J + 0.7 \\\\\n   \\textit{g-r $>$1.5 (r-i) + 0.8} & \\textit{r-i $>$1.5(i-Z) + 0.8} & \\textit{r-i $>$1.5(i-Z) + 0.8} & \\textit{i-z $>$2.0 (z-Y) + 1.1} & \u22120.5 $<$ Y \u2212 K $<$ 1.0 \\\\\n  \\cline{1-4}\n  \\multirow{3}{*}{\\textit{$i_{AB} > 22.2$}}  & \\multirow{3}{*}{\\textit{$z_{AB} > 23$}} & \\multirow{3}{*}{\\textit{$z_{AB} > 23$}}& \\multirow{3}{*}{\\textit{$z_{AB} > 22$}} & J \u2212 K $<$ 0.8 \\\\\n    &  &  & & undetected in \\textit{ugri} bands if available\\\\\n   \\hline\n \\end{tabular}\n \\begin{tablenotes}\n    \\item[a] \\cite{ono2018great}\n    \\item[b]  Our relaxed {\\it r} dropout criteria (see text for details.)\n    \\end{tablenotes}\n \\label{tab:dropouts_colour_criteria}\n  \\end{threeparttable}\n\\end{table*}\n\n\n\n\nTo find \\acp{HzRS}, we adopt the dropout criteria of \\cite{ono2018great}, shown in Table~\\ref{tab:dropouts_colour_criteria}.   The spectroscopically confirmed redshift ranges covered by each dropout, adopted from \\citet[Figure~6]{ono2018great}, are as follows: (i) g-dropout: $3\\le z \\le 4.5$ (ii) r-dropout: $4.3\\le z \\le 5.4$ (iii) i-dropout: $5.6\\le z \\le 6.2$.  To keep it simple, we use redshifts, $z\\sim4$, 5 and 6 to represent $g$, $r$, and $i$-band dropouts respectively. We use photometry from the KiDS DR4.1 multi-band catalogue (shown in Table~\\ref{tab:kiDS_filters}), based on the GAaP magnitudes corrected for both zero-point and Galactic extinction. The 5$\\sigma$ limiting magnitude for the $g$-band was used in the \\textit{$g - r$} colour if objects were undetected ( i.e. no entry in the catalogue) in $g$-band. Similarly, a 5$\\sigma$ limiting magnitude for the $r$ and $i$-bands were used to estimate \\textit{$r - i$} and \\textit{$i - z$} colours for $r$ and $i$ dropouts if objects were undetected in $r$ and $i$ band respectively.  In Table~\\ref{tab:total_rg-dropout}, we describe the photometric selection and number of sources remaining after applying $z \\sim 4$, 5, and 6 colour cuts. We present examples for all three dropouts, taken from our final sample, in Figure~\\ref{fig:example_dropouts}, showing their cutouts at each of the \\textit{ugri}ZYJHKs bands.\n\n Ideally, the signal-to-noise-ratio (SNR) of the sources should be used to \nmeasure\ntheir detection or non-detection in a given band. Since such information is not present in the KiDS catalogue, we utilized the mean 5$\\sigma$ limiting magnitude of each passband to define detection and undetection. Given that the 5$\\sigma$ limiting magnitude follows a continuous distribution \\citep{kuijken2019fourth}, this could  result in some dropouts being missed.\n\n\n\n\\begin{table*}\n    \\centering\n    \\begin{threeparttable}\n     \\caption{ Our initial sample: the number of radio sources remaining after each criterion. Sources lacking detection in a given dropout (no entry in the catalog) band is replaced with their respective mean 5$\\sigma$ limiting magnitude.}\n    \\begin{tabular}{c|c|c}\n    \\hline\n       Sample & Criteria & Radio source \\\\\n              & &  Count        \\\\\n       \\hline\n       G23-ASKAP &-- & 35,973 \\\\\n                                & cross-match with KiDS at 2\\,arcsec & 23\\,002 \\\\\n                                & After removing multiple matches & 17\\,447 \\\\\n       & imaflags\\_iso = 0 \\& &  \\multirow{2}{*}{17\\,396}  \\\\    \n       & nimaflags\\_iso = 0  & \\\\\n       & flag\\_gaap\\_$gri$ = 0  & 17\\,396 \\\\\n       & flag\\_gaap\\_$griZ$ = 0  & 17\\,376 \\\\\n                 \\hline\n        & col~1,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{4}{*}{229}\\\\\n        $z\\sim4$ sample  & Mag\\_gaap\\_i < 23.8 \\& & \\\\\n        ($g$ detected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                 \\hline\n         & col~1,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{4}{*}{6}\\\\\n                          & Mag\\_gaap\\_g = 25.12 \\& &\\\\\n         $z\\sim4$ sample  & Mag\\_gaap\\_i < 23.8 \\tnote{a} \\,\\& & \\\\\n         ($g$ undetected) &  Mag\\_gaap\\_u > 24.23 or undetected & \\\\\n                         \n                 \\hline\n         &   col~3,Table~\\ref{tab:dropouts_colour_criteria} colour cuts\\& & \\multirow{4}{*}{58}\\\\\n        $z\\sim5$ sample  & Mag\\_gaap\\_Z < 23.6 \\tnote{b} \\,\\& & \\\\\n        ($r$ detected)& Mag\\_gaap\\_u > 24.23 or undetected \\& & \\\\\n                      & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                 \\hline\n      &  col~4,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{5}{*}{6}\\\\\n        $z\\sim6$ sample  & Mag\\_gaap\\_Z < 23.6\\tnote{b} \\,\\& & \\\\\n        ($i$ detected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                      & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                      & Mag\\_gaap\\_r > 23.68 or undetected \\\\\n                 \\hline\n        & \n        col~4,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{5}{*}{9}\\\\\n        $z\\sim6$ sample  & Mag\\_gaap\\_Z < 23.6\\tnote{b} \\,\\& & \\\\\n        ($i$ undetected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                        & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                        & Mag\\_gaap\\_r > 23.68 or undetected \\\\\n                 \\hline         \n\n    \\end{tabular}\n    \\begin{tablenotes}\n    \\item[a]  $i$-band $5\\sigma$ limiting magnitude (AB) correspond to the end of the peak of the distribution \\citep[Figure~3]{kuijken2019fourth}.\n    \\item[b]  $Z$ band $5\\sigma$ limiting magnitude (AB) distribution correspond to the end of the peak of the distribution, assuming a spread of $\\pm0.5\\sigma$, given that the nominal $5\\sigma$ $Z_{lim} = 23.1$. \n    \\end{tablenotes}\n    \\label{tab:total_rg-dropout}\n    \\end{threeparttable}\n\\end{table*}\n\n\n\n\n\\begin{figure*}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=1.01\\textwidth]{g_dropout3.pdf}\n \\caption{An example of g dropout at $z\\sim4$ taken from our final sample:   J223541-311145. The size of the cutouts is $\\sim20\\,arcsec\\times15\\,arcsec$ and the circle is of radius $\\sim7.4\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=\\textwidth]{r_dropout2.pdf}\n \\caption{An example of r dropout at $z\\sim5$ taken from our final sample:   J230551-343338. The size of the cutouts is $\\sim24\\,arcsec\\times18\\,arcsec$ and the circle is of radius $\\sim8\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=\\textwidth]{i_dropout4.pdf}\n \\caption{An example of i dropout at $z\\sim6$ taken from our final sample:     J230246-293923. The size of the cutouts is $\\sim20\\,arcsec\\times15\\,arcsec$ and the circle is of radius $\\sim6\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\n\\caption{KiDS-\\textit{ugri} and ViKING-ZYJHKs images of Lyman dropouts at each redshift are illustrated above in the order of increasing wavelength. The Lyman break can be identified by combining photometry in three consecutive bands. At $z \\sim 4$, the Lyman limit ($\\lambda\\textsubscript{rest}=912\\textup{\\AA}$) falls in the g-band and shorter wavelengths are significantly absorbed, as indicated by faint $u$ and $g$ bands. Similarly for $z \\sim 5$ and 6 sources, the Lyman limit falls in the $r$ and $i$-bands respectively.}\n\\label{fig:example_dropouts}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[trim={0 370 0 25 clip=true},width=\\textwidth]{z_dropout3.pdf}\n    \\caption{VIKING bands of a Z dropout at $z\\sim7$ taken from our final sample.  The size of the cutouts is $\\sim25\\,arcsec\\times20\\,arcsec$ and the circle is of radius $\\sim8\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source. }\n    \\label{fig:my_label}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsubsection{Removing low-$z$ interlopers from our $z \\gtrsim 4-6$ sample}\n\\label{sec:interloper1}\n\nLow-$z$ objects such as dwarf stars and passive galaxies can enter the selection region defined by Lyman dropout colour cuts due to photometric errors, even though intrinsically they do not enter the colour selection window. We evaluate the contamination rate in the $z \\sim 4$, 5 and 6 selection window by testing the Lyman dropout technique on a set of objects with known spectroscopic redshifts. We chose the COSMOS field \\citep{scoville2007cosmic}, a well-studied region of the sky where both broadband optical photometry and spectroscopic redshifts are available.  We used the following catalogues of the COSMOS field, available in the public \\ac{IRSA} domain, to test $g$, and $r \\&$ $i$-band dropout techniques respectively ; (i) COSMOS Photometry Catalogue January 2006 (hereafter, COSMOS2006 catalog; \\cite{capak2007first})  and (ii) COSMOS2015 Catalog \\citep{laigle2016cosmos2015}. The spectroscopy for COSMOS sources was obtained from COSMOS DEIMOS Catalogue \\citep{hasinger2018deimos}.  We utilized the  Subaru-\\textit{griz} photometry in the COSMOS catalogues to test the colour cuts.  Furthermore, in this paper, we follow lowercase and uppercase notation in the literature for the Subaru {\\it z} filter and the VIKING Z filter respectively.\n\n We crossmatched the COSMOS 2006 catalogue and the COSMOS DEIMOS Catalogue at 1\\,arcsec, resulting in 8906 matches. We further excluded multiple matches and apply the following criteria as recommended in \\cite{capak2007first} to obtain a cleanest sample: blend$\\_$mask = 0, i$\\_$mask = 0, b$\\_$mask = 0, and v$\\_$mask = 0. This results in 6\\,787 unique sources that have been deblended and are without any photometric flags to test the $g$-band dropout colour cuts. We further corrected the $g$, $r$, and $i$ magnitudes for Galactic extinction following the recommendations in \\cite{capak2007first}.\n\n Similarly, we crossmatched the COSMOS2015 catalogue and the COSMOS DEIMOS Catalogue at 1\\,arcsec, resulting 7640 matches. The sources with multiple matches were excluded and the following criteria were applied as per \\cite{laigle2016cosmos2015}: $flag\\_HJMCC=0 \\& flag\\_cosmos=1 \\& flag\\_peter=0$, giving a source count of 7\\,502. We finally applied respective Galactic extinction corrections to $r$, $i$, $z$, and $Y$ bands as per \\cite{laigle2016cosmos2015}. \n\nTable~\\ref{tab:dropouts_colour_criteria} shows that $z \\sim 4$, 5, and 6 galaxy candidates can be selected based on their \\textit{gri}, \\textit{riz}, \\textit{izy} colours respectively. We demonstrate this in Figure~\\ref{fig:cc_plot_cosmos} by plotting the spectroscopic redshift distribution of COSMOS sources in \\textit{g$-$r} vs. \\textit{r$-$i}, \\textit{r$-$i} vs. \\textit{i$-$z} and \\textit{i$-$z} vs. \\textit{z$-$Y} colour-colour space. \n To test the $z \\sim 7$ or Z-band dropout colour cuts, deep spectroscopic data ($z_{spec}>6.4$) is needed, which is not available in the COSMOS DEIMOS catalog. It is evident that the photometric selection window of g-dropouts (the black box) encompasses almost all $z \\sim 4$ sources, with a small contribution from low-$z$ ``interloper'' sources. By contrast, the r-dropout selection window misses a significant fraction of $z \\sim 5$ sources.   Therefore, we relaxed the $r$ dropout colour cuts as follows:\n\n\\begin{equation} \\label{eq:1}\n r-i > 1.0; \\\\\ni-z < 0.75 ; \\\\\nr-i > 1.5*(i-z)+0.8\n\\end{equation}\n\nThe resulting colour locus is indicated by black dashed lines, showing \n that more $z \\sim 5$ sources get included than low-$z$ ones.  \n\n\nIn Figure~\\ref{fig:mag_redshift_cosmos}, we demonstrate that $g$ and $r$ dropouts suffer significant contamination from low-$z$ sources at the bright end. Based on this, we introduce further selection criteria in the form of a magnitude cut-off in $i$-band (i\\textsubscript{AB} $>$ 22.2) for $g$-dropouts and in $z$-band  (z\\textsubscript{AB} $>$ 23) for $r$-dropouts, which helps to reduce the contribution from low-z sources (see bottom row of Table~\\ref{tab:dropouts_colour_criteria}). \n\n We further note that our sample size of $i$-band dropouts is \ntoo small\nto draw any conclusion. The $Z$ band magnitude cut-off ($Z_{AB}>24$) implied by the figure~\\ref{fig:mag_redshift_cosmos} is too high and the VIKING data is also not deep enough to apply this cut-off.  Therefore, we performed a literature search to to find the $Z$-band magnitude of known $z\\sim6$ quasars. Based on the spectroscopically confirmed $i$ - band dropouts in \\citet[Table~3]{venemans2015first}, we applied a magnitude cut-off, Z\\textsubscript{AB} $>$ 22 for $i$-dropouts.  \n\n\n\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{gir_cc_cosmos2.png} \n\\end{subfigure}\\hspace{0.5cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{riz_cc_cosmos2.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{izy_cc_cosmos2.png} \n\\end{subfigure}\n\\caption{\\textbf{(Top Left)} \nSelection of $z\\sim4$ sources by means of $g$-band dropout technique using $gri$ broadband filters: \\textit{(g-r)} vs. \\textit{(r-i)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box represents the $g$ dropout selection criteria in \\protect\\cite{ono2018great}. It encompasses almost all $z\\sim4$ sources, but with a small contamination from low-$z$ sources too. \\textbf{(Top Right)} Selection of $z\\sim5$ sources by means of $r$-band dropout technique using $riz$ broadband filters: \\textit{(r-i)} vs. \\textit{(i-z)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box (solid lines) represents the $r$ dropout selection criteria in \\protect\\cite{ono2018great}.  The black box bordered by dashed black lines represent the colour locus from relaxed $r$ band colour-cuts. \n \\textbf{(Bottom)} Selection of $z\\sim6$ sources by means of $i$-band dropout technique using $izY$ broadband filters: \\textit{(i-z)} vs. \\textit{(z-Y)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box represents the $i$ dropout selection criteria in \\protect\\cite{ono2018great}. }\n\n\\label{fig:cc_plot_cosmos}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{imag_zspec_final.png} \n\\end{subfigure}\\hspace{0.5cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{zmag_zspec_z5_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{zmag_zspec_z6_final.png} \n\\end{subfigure}\n\\caption{Selecting low-$z$ interlopers in $g$ and $r$ dropout samples in COSMOS field by utilising the respective detection bands of each dropout: \\textbf{(Top Left)} $i$ magnitude of $g$ dropouts together with their spectroscopic redshifts. \\textbf{(Top Right)} $z$ magnitude of $r$ dropouts together with their spectroscopic redshifts.   Blue solid circles represent the $r$ dropouts selected using \\protect\\cite{ono2018great} criteria and red crosses, the $r$ dropouts selected using our relaxed colour-cuts (Equation~\\ref{eq:1}).    (Bottom:) $z$ magnitude of $i$ dropouts together with their spectroscopic redshifts. } \n\\label{fig:mag_redshift_cosmos}\n\\end{figure*}\n\\begin{figure*}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{g_dropout_final.png}\n\\end{subfigure} \\hspace{1 cm}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{r_dropout_final.png}\n\\end{subfigure}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{i_dropout_final.png}\n\\end{subfigure}\\hspace{1 cm}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{z_dropout_final.png}\n\\end{subfigure}\n\\caption{ Colour-colour diagram of our final sample of \\acp{HzRS} at $z\\gtrsim$4-7 selected via the Lyman dropout technique. The black box shows the selection criteria of $g$, $r$, $i$, and $Z$ dropouts in $gir$, $riZ$, $iZY$, and $ZYJ$ colour-colour plane respectively.}\n\\label{fig:cc_dropouts}\n\\end{figure*}\n\n\n\n\n\n\\subsection{Selection of radio sources at $z\\sim7$}\n\n\nWe utilised VIKING photometry to search for radio sources in the redshift range, $6.44\\le z \\le 7.44$. \\cite{venemans2013discovery} demonstrated that Lyman dropouts at $z\\sim7$ (a.k.a Z dropouts) can be selected using  ZYJ near-IR filters. We cross-matched G23-ASKAP radio catalogue and VIKING DR5 catalogue at a search radius of 2$\\,arcsec$,  which  gives a false ID rate of 7.9\\% . Using the selection method in \\cite{venemans2013discovery} (see Table~\\ref{tab:dropouts_colour_criteria}), we select a sample of 3 radio source candidates at $z\\sim7$, of which one is a known radio-quasar, VIK J2318\u22123113, at $z=6.44$ \\citep{ighina2021radio}. Scattering of foreground galaxies into the Z-dropout selection region in the ZYJ color-color space is minimised by selecting point sources only by applying the criterion: pGalaxy$<0.95$ ( see \\cite{venemans2013discovery} and reference therein for details), where pGalaxy is the probability that the source is a galaxy.   \n\n\\subsection{Estimate of Reliability}\n\\label{reliability}\nFinally, we estimate the success rate of our magnitude cutoff by counting the fraction of low-$z$ sources remaining after applying the $i$-band and $z$-band cutoff to $g$ and $r$ \\& $i$ dropouts respectively. \n A sample of  294 sources were identified from the COSMOS catalog as satisfying the $z \\sim 4$ colour cuts (Table~\\ref{tab:dropouts_colour_criteria}), of which  75 sources have $z_\\textrm{spec}$ below 3.0, resulting in a contamination rate of 25.5\\%. The further application of the i\\textsubscript{AB} $>$ 22.2 criterion reduces the low-z source count to 29 and the total sample size to 248. This gives a final contamination rate of  11.7\\%.\n\n A total of 78 sources were identified as $r$ dropouts using \\cite{ono2018great} criteria, \nof which 19\nhave $z_\\textrm{spec} < 4.3$ giving a contamination rate of $\\sim24.4\\%$. On the other hand, relaxed $r$ band colour cuts (Equation~\\ref{eq:1}) selected 127 sources in total, of which 29 have $z_\\textrm{spec} < 4.3$ resulting a contamination rate of $\\sim22.8\\%$. This shows that relaxed $r$ band colour cuts select $\\sim1.7$ times more $z_\\textrm{spec} > 4.3$ sources than the \\cite{ono2018great} criteria while the contamination rate remains almost constant. We further reduce the contamination rate to 19\\% by applying a  z\\textsubscript{AB} $>$ 23 cutoff. \n\n Due to the dearth of deep spectroscopic data, only 14 sources were selected as $i$ dropouts, 6 of which have a $z_\\textrm{spec}<5.6$ classifying as interlopers. This gives an initial contaminant rate of $\\sim 43 \\%$ and the further application of magnitude cut-off, $Z_{AB} > 22$, results a final contaminants rate of $\\sim 38.4\\%$.\n\n This shows that the $z \\sim 4$ sample is  $\\sim88\\%$ reliable while the $z \\sim 5$ sample is around 81\\% reliable. \nThe $z \\sim 6$ sample is $\\sim62\\%$ reliable, which is likely to be a lower limit given that the sample size was smaller.\nThere is no robust way of estimating the reliability for the $z\\sim7$ sample, as there is a lack of deep spectroscopic data in COSMOS. \n\n We applied these magnitude cut-offs to the sources in Table~\\ref{tab:total_rg-dropout} to select our final sample of radio source candidates at $z\\gtrsim4-6$ in the G23-\\ac{ASKAP} field. The resulting source count of each dropout is shown in Table~\\ref{tab:final}. As a further check on reliability, we cross-matched our sample with the GAMA spectroscopic redshift \\citep{baldry18} catalogue and the Gaia Early Data Release 3 \\citep[EDR3,][]{brown2020gaia}. The Gaia satellite measures parallaxes and proper motions for nearby stars in our Milky Way, and allow us to identify cool dwarf stars, if any, in our final sample. Gaia EDR3 is the latest release, providing information for about 1.8 billion objects.\n\nNone of our sources were found to have either GAMA spectroscopic or GAIA counterparts out to a search radius of 4~arcsec. The lack of GAMA spectroscopic counterparts is consistent with a high redshift, as the GAMA spectroscopic survey is complete to $z\\sim0.4$. Similarly, the lack of GAIA counterparts confirms that (i) no Milky Way stars are present in our sample, and (ii) no low-z quasars that are bright enough to be detected by GAIA are in our final sample.\n\n\n\n\\section{Results}\n\\label{sec:result}\n\n\n \n  Using the Lyman Dropout photometric technique and our additional magnitude cut-offs (Table~\\ref{tab:dropouts_colour_criteria}) to reduce low-$z$ interlopers, we select our final sample of 148 radio source candidates at $z \\gtrsim 4-7$, where (i) {\\it g} \\& {\\it i} dropouts criteria come from \\cite{ono2018great} (ii) {\\it r } dropout criteria from this study (Equation~\\ref{eq:1}) and (iii) Z dropout criteria from \\cite{venemans2013discovery}. \n \n  The colour-colour plot of our final sample is shown in Figure~\\ref{fig:cc_dropouts}. Furthermore, we identify a known radio quasar at $z=6.44$ and 2 radio source candidates at $z\\sim7$ using the Z-dropout selection technique from \\cite{venemans2013discovery}.  Thus our final sample of 149 radio sources in 50\\,deg\\textsuperscript{2} implies a sky density of $\\sim3$ per $deg^{2}$ for $S_{888}\\gtrsim0.1$\\,mJy radio sources at $z\\gtrsim4$. For comparison, \\cite{norris21} suggested a sky density of $\\sim5$ per $deg^{2}$ beyond $z\\sim4$ at EMU flux limit,\n based on simulations.  We present the catalogues of $z \\gtrsim 4-6$ and $z\\sim7$ sources, including KiDS\/VIKING and WISE W1 photometry, in Table~\\ref{tab:catalog} and Table~\\ref{tab:z7_sample} respectively.\n \n\n\n\n\n\\begin{table}\n \\caption{Our final sample of  $ z\\gtrsim4-7$ candidates in G23 field selected using Lyman dropout colour cuts (Table~\\ref{tab:dropouts_colour_criteria}) together with our magnitude cutoffs from Figure~\\ref{fig:mag_redshift_cosmos}}.\n \\centering\n \\begin{tabular}{c|c}\n \\hline\n  Redshift ($z$) & Sample Size \\\\\n  \\hline\n  $z\\sim4$ &  117\\\\\n  $z\\sim5$ &  14 \\\\\n  $z\\sim6$ &  15\\\\\n  $z\\sim7$ & 3\\\\\n  \\hline\n \\end{tabular}\n \\label{tab:final}\n\\end{table}\n\n\n\n\\clearpage\n\\onecolumn\n\\begin{longtable}[c]{ccccccccccc}\n \\caption{Our final $z\\gtrsim4-6$ sample, showing radio properties from \\ac{ASKAP} and optical\/ IR photometry from KiDS and CATWISE catalogues respectively. Missing optical magnitudes indicate non-detection in that respective filter. The missing WISE magnitude (W1) indicate no counterpart within 2~arcsec of the KiDS position.   KiDS\/VIKING magnitudes are given in the AB system and the WISE magnitudes in the Vega unit. }\\\\\n \\hline\n Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5}&Mag\\_u &Mag\\_g& Mag\\_r & Mag\\_i & Mag\\_Z & W1 \\\\\n  & name & (deg) & (deg) & (mJy) & & & & & & (mag)  \\\\\n  \\hline\n \\endfirsthead\n \\hline\n  Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5}&Mag\\_u &Mag\\_g& Mag\\_r & Mag\\_i & Mag\\_Z & W1 \\\\\n  & name & (deg) & (deg) & (mJy) & & & & & & (mag)  \\\\\n  \\hline\n  \\endhead\n  \\multicolumn{11}{c}{$z\\sim4$ Sample ($g$ dropouts) } \\\\\n  \\hline\n\n1&  J225745-311209 & 344.437  & -31.203   & 0.32     & 25.067 & 25.779  & 24.108  & 23.549  & 22.928  &        \\\\\n2&  J225802-345205 & 344.511  & -34.868    & 0.948    &            & 25.954  & 23.293  & 22.808  & 21.720  & 17.588 \\\\\n3&  J225835-325509 & 344.649  & -32.919   & 0.237    &            & 25.220  & 23.268  & 22.689  & 21.573  & 15.633 \\\\\n4&  J225915-314116 & 344.815   & -31.688   & 0.57     &            & 24.348  & 22.954  & 22.649  & 21.796  & 16.055 \\\\\n5&  J230041-322125 & 345.172  & -32.357   & 0.247    &            & 25.854  & 23.887  & 23.248  & 21.810  & 15.95  \\\\\n6&  J230122-294717 & 345.342  & -29.788   & 0.254    & 24.800 & 25.084   & 23.678  & 23.685  & 23.143  &        \\\\\n7&  J230156-331615 & 345.484  & -33.271   & 0.261    & 24.993  & 25.791  & 23.483  & 22.506  & 21.295   & 15.556 \\\\\n8&  J230223-294300 & 345.599  & -29.716   & 0.584    & 24.999  & 25.341   & 23.511  & 22.884   & 22.187 & 17.686 \\\\\n9&  J230234-335010 & 345.645  & -33.836   & 0.246    &            & 25.306  & 23.750 & 23.314  & 23.198 &        \\\\\n10&  J230249-322437 & 345.708  & -32.411   & 0.233    &            & 26.273  & 24.163   & 23.632   & 22.896  & 16.641 \\\\\n11&  J230306-333335 & 345.778  & -33.559   & 0.282    &            &            & 23.845  & 23.724   &            &        \\\\\n12&  J230317-345556 & 345.824  & -34.932   & 2.428    &            & 25.179  & 23.003 & 22.217 & 21.372 & 16.771 \\\\\n13&  J230432-341722 & 346.133   & -34.289   & 6.517    & 25.132 & 24.487  & 23.360   & 23.481  & 23.491  &        \\\\\n14&  J230504-292342 & 346.267  & -29.395   & 0.646    &            & 25.831  & 23.852  & 23.297  & 22.074  & 17.746 \\\\\n15&  J230516-331612 & 346.317  & -33.270   & 0.421    &            & 25.096  & 23.024   & 22.381   & 22.067   &        \\\\\n16&  J230544-351610 & 346.434  & -35.269   & 0.302    &            & 24.140  & 23.057  & 23.002  & 22.886  &        \\\\\n17&  J230547-293327 & 346.449  & -29.557   & 2.886    & 24.393   &            & 23.511  & 23.343  & 22.727   &        \\\\\n18&  J230718-310125 & 346.826   & -31.024   & 0.364    &            & 24.013  & 22.692  & 22.784  & 22.728  &        \\\\\n19&  J230831-350015 & 347.133  & -35.004   & 0.203    &            & 24.927  & 23.111   & 22.451  & 21.681  & 16.86  \\\\\n20&  J230832-340644 & 347.137   & -34.112   & 8.009    & 25.594  & 24.677  & 23.069  & 22.766  & 22.104  & 16.219 \\\\\n21&  J230905-334318 & 347.272  & -33.722   & 0.204    &            & 25.661   & 23.539  & 22.862  & 21.849  &        \\\\\n22&  J230937-323421 & 347.405  & -32.573   & 0.272    &            & 25.599   & 23.829  & 23.760   & 22.196  & 16.971 \\\\\n23&  J230940-335143 & 347.418  & -33.862     & 0.298    &            & 24.765  & 23.522  & 23.691  & 24.487  &        \\\\\n24&  J230942-335049 & 347.427  & -33.847   & 0.304    &            &            & 23.385  & 22.792  & 21.804  & 16.54  \\\\\n25&  J231057-294135 & 347.740   & -29.693   & 0.431    &            & 25.159  & 23.852  & 23.621  & 22.309  & 16.91  \\\\\n26&  J231117-323952 & 347.822   & -32.664   & 0.247    &            & 25.543  & 24.052 & 23.674  & 23.917  &        \\\\\n27&  J231148-311231 & 347.950  & -31.209   & 0.323    & 25.134  & 24.726  & 22.381  & 22.753  & 22.904  & 18.202 \\\\\n28&  J231149-304758 & 347.958  & -30.799   & 0.292    & 24.414   & 24.550   & 23.228   & 23.087  & 22.582   & 16.862 \\\\\n29&  J231210-332436 & 348.045  & -33.410   & 0.26     & 24.598  & 25.670  & 24.111  & 23.718  & 23.112  & 17.172 \\\\\n30&  J231421-344141 & 348.587   & -34.695   & 0.749    & 24.784   & 24.396  & 22.718  & 22.233  & 22.000  & 16.423 \\\\\n31&  J231444-291949 & 348.684  & -29.330   & 0.276    &            & 26.165   & 23.711  & 22.739  & 21.711  & 16.301 \\\\\n32&  J231449-293938 & 348.706  & -29.661   & 1.205    & 24.960  & 25.635  & 23.636  & 23.097  & 21.851  & 16.812 \\\\\n33&  J231508-312105 & 348.784  & -31.351   & 0.244    &            & 25.858  & 24.062  & 23.455  & 22.462   & 17.192 \\\\\n34&  J231508-341955 & 348.787  & -34.332    & 98.088   &            & 25.778   & 23.454   & 22.511   & 21.470  & 15.897 \\\\\n35&  J231555-311458 & 348.979  & -31.249   & 0.472    &            & 26.288  & 23.857  & 23.423   & 22.709   & 17.889 \\\\\n36&  J231604-324740 & 349.020  & -32.795   & 0.151    &            & 26.329  & 24.171  & 23.403  & 22.477  & 16.847 \\\\\n37&  J231617-303200 & 349.073  & -30.533   & 3.779    &            & 25.763  & 23.302   & 22.391  & 21.255  & 15.878 \\\\\n38&  J231632-331953 & 349.134  & -33.332   & 1.342    & 24.946   & 24.309  & 22.669  & 22.263  & 21.888  & 17.111 \\\\\n39&  J231645-301948 & 349.189  & -30.330  & 0.763    &            & 24.803  & 23.166  & 22.959  & 21.994   & 18.09  \\\\\n40&  J231648-303629 & 349.201  & -30.608   & 0.356    &            & 25.818  & 23.713  & 23.062  & 22.658  & 16.939 \\\\\n41&  J231719-315344 & 349.332  & -31.896   & 0.205    & 25.291  & 24.586  & 23.584  & 23.451    & 23.077   &        \\\\\n42&  J231723-301556 & 349.348  & -30.266   & 0.464    &            & 24.145   & 22.748  & 22.487  & 22.705  & 18.284 \\\\\n43&  J231737-313228 & 349.407  & -31.541   & 0.259    & 24.763  & 26.235  & 23.565  & 22.605   & 21.880   & 16.663 \\\\\n44&  J231752-311151 & 349.468  & -31.197   & 0.324    &            & 24.784  & 23.772  & 23.705  & 22.739  & 16.736 \\\\\n45&  J231826-323746 & 349.610  & -32.629    & 0.374    & 24.867  & 25.198  & 23.653  & 23.676  & 22.280  & 16.01  \\\\\n46&  J231828-310408 & 349.618  & -31.069   & 0.197    &            & 24.853  & 23.426   & 23.403  & 22.139  & 17.125 \\\\\n47&  J231850-303818 & 349.708  & -30.638   & 0.2      &            & 25.686  & 22.949  & 22.236  & 21.583  & 16.557 \\\\\n48&  J231853-293420 & 349.722  & -29.572   & 0.361    & 24.365   & 26.231  & 24.235  & 23.583  & 22.366  & 16.635 \\\\\n49&  J232019-294205 & 350.081  & -29.702  & 0.572    &            & 25.674  & 24.177  & 23.777  & 22.864  & 15.97  \\\\\n50&  J232020-343818 & 350.084  & -34.638   & 0.257    &            & 25.217  & 23.124  & 22.267  & 21.1660  & 16.308 \\\\\n51&  J232034-305242 & 350.143  & -30.879   & 0.506    &            & 26.007  & 23.508  & 22.591  & 21.225   & 16.13  \\\\\n52&  J232043-320457 & 350.182  & -32.083   & 3.006    & 25.217  & 25.887  & 23.793   & 23.129  & 22.256  & 16.822 \\\\\n53&  J232130-320208 & 350.377  & -32.036   & 0.294    & 25.261  & 24.562  & 23.421  & 23.215  & 21.940  & 17.03  \\\\\n54&  J232135-320117 & 350.399  & -32.022   & 0.266    &            & 24.778   & 22.919   & 22.309  & 21.839  & 16.359 \\\\\n55&  J232140-311522 & 350.417  & -31.256   & 0.47     &            & 25.638  & 23.374  & 22.705   & 21.997  & 17.304 \\\\\n56&  J232246-331448 & 350.691  & -33.247     & 0.277    & 24.752  & 26.199  & 23.584  & 22.589  & 21.546  & 16.676 \\\\\n57&  J232331-345632 & 350.882  & -34.942   & 0.639    &            & 26.101  & 23.673  & 22.688  & 21.728  & 15.694 \\\\\n58&  J232338-350353 & 350.910  & -35.065   & 0.988    &            & 24.471   & 23.451  & 23.767  & 22.534  & 18.065 \\\\\n59&  J232347-344625 & 350.948   & -34.774   & 0.726    &            & 25.682  & 23.363   & 22.491  & 21.452   & 15.994 \\\\\n60&  J232413-303039 & 351.057  & -30.511   & 0.457    &            & 26.281  & 23.783   & 22.797  & 21.864  & 16.669 \\\\\n61&  J232515-295957 & 351.314  & -29.999   & 1.317    &            & 24.565  & 22.944  & 22.509   & 22.097  & 17.487 \\\\\n62&  J232515-314448 & 351.314  & -31.747   & 1.431    & 24.848  & 26.250  & 23.447  & 22.489  & 21.277  & 16.122 \\\\\n63& J222915-334311  & 337.314  & -33.719   & 1.992    &            & 25.057  & 23.829   & 23.792   & 23.329    &        \\\\\n64& J223018-304213  & 337.576  & -30.704   & 2.954    &            & 25.314  & 23.629  & 23.089   & 22.573  & 17.405 \\\\\n65& J223054-322552  & 337.727 & -32.431    & 75.829   &            & 25.564  & 23.444  & 22.627  & 21.861  & 17.538 \\\\\n66& J223215-301935  & 338.066   & -30.327   & 10.809   &            & 26.243  & 23.488   & 23.135   & 22.872  & 18.29  \\\\\n67& J223508-295809  & 338.785  & -29.969   & 1.081    &            & 24.139  & 22.897   & 22.607  & 22.438  &        \\\\\n68& J223531-291912  & 338.879  & -29.320   & 0.744    & 24.549  & 25.323   & 23.572  & 23.050   & 22.325   & 16.968 \\\\\n69& J223533-343305  & 338.889  & -34.552   & 1.942    & 24.406  & 24.286    & 23.031  & 22.894   & 21.702  & 17.279 \\\\\n70& J223541-311145  & 338.922  & -31.196   & 0.689    &            & 26.473  & 24.012  & 23.065  & 21.906  & 16.757 \\\\\n71& J223710-305549  & 339.294  & -30.930   & 2.54     &            & 25.052  & 23.322  & 23.006  & 22.722  & 16.801 \\\\\n72& J223729-325838  & 339.374  & -32.977  & 0.337    & 24.749  & 25.078   & 23.796  & 23.485  & 22.158   & 16.236 \\\\\n73& J223743-333305  & 339.432  & -33.551   & 1.05     &            & 24.747  & 23.514   & 23.384   & 22.600  & 16.724 \\\\\n74& J223831-330728  & 339.629  & -33.124   & 0.249    &            & 25.445  & 23.981  & 23.727  & 24.024   &        \\\\\n75& J223936-295303  & 339.900  & -29.884    & 2.703    &            & 25.995  & 23.209 & 22.222  & 21.617   & 16.493 \\\\\n76& J224017-325128  & 340.073  & -32.858   & 0.398    & 24.662  & 25.188  & 23.179  & 22.451   & 22.441 & 16.653 \\\\\n77& J224040-295600  & 340.168  & -29.933    & 2.673    & 24.283  & 24.770  & 22.952   & 22.466  & 20.989  & 16.221 \\\\\n78& J224047-331331  & 340.197  & -33.225   & 0.646    &            & 25.308  & 23.958  & 23.689   & 23.193  &        \\\\\n79& J224138-331346  & 340.411  & -33.229    & 0.242    &            & 25.162  & 23.246  & 22.570  & 21.842  & 17.368 \\\\\n80& J224145-340622  & 340.439   & -34.106    & 66.186   &            & 26.2801  & 23.891   & 22.992  & 21.688  & 16.254 \\\\\n81& J224203-333606  & 340.513  & -33.602   & 0.769    & 25.436   & 25.747  & 23.598  & 23.154  & 24.637   & 16.188 \\\\\n82& J224246-335928  & 340.692  & -33.991   & 1.751    &            & 25.566  & 23.892  & 23.647  & 23.481  &        \\\\\n83& J224249-333814  & 340.706  & -33.637   & 0.55     &            & 25.751  & 23.515   & 22.745  & 21.976  & 17.565 \\\\\n84& J224308-313622  & 340.784  & -31.606   & 0.662    & 24.432 & 25.657  & 24.005  & 23.593  & 22.933  & 17.425 \\\\\n85& J224311-332250  & 340.796  & -33.381   & 4.432    &            & 25.643  & 23.491  & 22.924  & 21.404   & 16.056 \\\\\n86& J224354-333900  & 340.976  & -33.650   & 0.272    & 25.384  & 25.028  & 23.303   & 23.028  & 21.557  & 16.502 \\\\\n87& J224401-315431  & 341.0056   & -31.909   & 11.74    &            & 26.160  & 23.101   & 22.309  & 21.1800  & 15.864 \\\\\n88& J224417-350540  & 341.072  & -35.094   & 0.622    & 24.869  & 25.819  & 23.907  & 23.741  & 22.267   & 17.523 \\\\\n89& J224445-314547  & 341.188 & -31.763   & 4.325    &            & 25.334  & 23.252     & 22.967  & 21.969  &        \\\\\n90& J224519-342942  & 341.329  & -34.495   & 0.535    &            & 25.644  & 23.446  & 22.529  & 21.990  & 16.562 \\\\\n91& J224540-313747  & 341.418  & -31.629   & 0.221    & 25.059  & 24.642  & 22.851   & 22.319  & 21.532  & 16.265 \\\\\n92& J224540-330700  & 341.419  & -33.117   & 1.679    & 24.517   & 25.802  & 23.269  & 22.568  & 21.588   & 16.067 \\\\\n93& J224552-312733  & 341.468  & -31.4592   & 0.409    &            & 26.263  & 23.615  & 22.636  & 22.315   & 16.989 \\\\\n94& J224601-310331  & 341.508 & -31.059   & 0.685    & 25.250  & 26.278  & 24.116  & 23.278  & 22.031  & 16.936 \\\\\n95& J224615-285257  & 341.565  & -28.883   & 1.177    &            & 24.328  & 22.526  & 22.237  & 22.035  &        \\\\\n96& J224623-311641  & 341.597  & -31.278     & 0.206    & 24.845   & 25.129  & 23.845  & 23.735  & 22.802  & 18.402 \\\\\n97& J224629-323825  & 341.622 & -32.640  & 0.824    &            & 25.533  & 23.647   & 23.141  & 22.331  & 16.925 \\\\\n98& J224630-343853  & 341.628  & -34.648   & 0.599    & 24.742  & 26.381  & 24.066   & 23.071   & 21.513  & 16.402 \\\\\n99& J224642-311527  & 341.677 & -31.258   & 0.189    & 24.730  & 25.230   & 23.807   & 23.410  & 22.712  &        \\\\\n100& J224710-321939  & 341.792   & -32.328   & 0.476    & 24.249  &            & 23.011   & 22.249   & 21.336  & 16.425 \\\\\n101& J224716-310058  & 341.817  & -31.016   & 0.32     &            & 26.009  & 23.856  & 23.021  & 21.859  & 17.402 \\\\\n102& J224812-335035  & 342.051  & -33.843   & 0.36     & 25.272  & 26.052  & 24.037  & 23.753  & 21.790  & 16.874 \\\\\n103& J224918-314457  & 342.327  & -31.749   & 4.265    &            & 25.877  & 23.815  & 23.540  & 24.035  &        \\\\\n104& J224955-294536  & 342.479  & -29.760     & 0.294    & 24.707  & 26.023  & 23.892  & 23.036  & 22.479  & 16.346 \\\\\n105& J225029-350452  & 342.622  & -35.0811   & 0.663    &            & 25.287  & 23.936  & 23.575  & 22.385  & 16.907 \\\\\n106& J225052-331301  & 342.719  & -33.217   & 0.49     & 24.649  & 25.228  & 23.448  & 22.800  & 21.656  & 16.539 \\\\\n107& J225214-310846  & 343.061  & -31.146   & 2.607    &            & 26.549  & 24.020   & 23.083  & 22.204  & 16.635 \\\\\n108& J225240-302302  & 343.166  & -30.384   & 0.384    &            & 26.053  & 23.786   & 23.082  & 21.805  & 16.802 \\\\\n109& J225257-351113  & 343.238  & -35.187   & 0.275    & 25.510  & 25.626  & 23.263  & 22.320  & 21.177  & 16.124 \\\\\n110& J225300-302322  & 343.253  & -30.389   & 0.49     &            & 24.503  & 23.148  & 22.869  & 22.856  &        \\\\\n111& J225312-305344  & 343.303  & -30.896   & 0.188    & 24.612  & 25.492  & 23.359  & 22.624 & 22.381  & 16.758 \\\\\n112& J225314-295139  & 343.311  & -29.861   & 0.365    &            & 25.334  & 23.804  & 23.712  & 24.634  &        \\\\\n113& J225332-313319  & 343.386  & -31.555   & 0.242    & 24.613  & 25.953  & 23.599   & 23.023   & 21.794  & 16.304 \\\\\n114& J225343-313305  & 343.429  & -31.552  & 0.367    &            & 25.312  & 23.191  & 22.481  & 21.788   & 16.938 \\\\\n115& J225647-285027  & 344.196  & -28.841   & 2.049    &            & 24.798  & 23.615   & 23.382   & 22.680  & 17.25  \\\\\n116& J225733-313857  & 344.388  & -31.649   & 0.249    &            & 25.377  & 23.456  & 22.837  & 21.364  & 16.001 \\\\\n117& J225827-342715  & 344.614  & -34.454   & 0.286    & 25.139  & 26.231  & 24.271  & 23.650  & 22.228   & 16.669 \\\\\n\n\n\\hline\n\\multicolumn{11}{c}{$z\\sim5$ Sample  ($r$ dropouts)} \\\\\n\\hline\n\n1&  J231423-331509 & 348.596  & -33.253    & 0.232    &            & 26.248  & 24.738 & 23.642  & 23.575  & 16.47  \\\\\n2& J224105-345956  & 340.271  & -34.999   & 0.261    &            & 25.664  & 24.804  & 23.343  & 23.205  & 16.581 \\\\\n3&  J232503-340057 & 351.264  & -34.016    & 0.293    & 25.392  &            & 24.578 & 23.471  & 23.267  & 17.989 \\\\\n4&  J231919-320058 & 349.831  & -32.016  & 0.294    &            &            & 25.372  & 23.793  & 23.308  & 17.586 \\\\\n5& J224820-301317  & 342.086  & -30.221    & 0.295    & 24.808  & 25.333  & 24.632   & 23.402    & 23.171 & 17.793 \\\\\n6&  J230855-335352 & 347.229  & -33.898   & 0.323    &            &            & 24.749  & 23.372  & 23.063 & 17.508 \\\\\n7& J224858-343550  & 342.242  & -34.597   & 0.356    & 25.101   & 26.083  & 24.681  & 23.457  & 23.569     &        \\\\\n8&  J232021-315838 & 350.090  & -31.977   & 0.382    &            & 25.374  & 24.706  & 23.483   & 23.327  & 17.392 \\\\\n9&  J230235-292456 & 345.648  & -29.415    & 0.443    &            & 25.676  & 24.440  & 23.003 & 23.181   & 16.789 \\\\\n10&  J232035-314851 & 350.149  & -31.814   & 0.453    & 25.137  & 25.209  & 24.388  & 23.235    & 23.167  &        \\\\\n11& J225340-300626  & 343.417  & -30.107   & 0.604    & 24.703  & 26.143  & 24.253   & 23.164  & 23.186  & 16.329 \\\\\n12&  J230551-343338 & 346.463  & -34.561   & 1.248    &            & 26.491  & 25.365  & 23.768  & 23.568   & 15.901 \\\\\n13&  J230946-314050 & 347.446  & -31.681   & 1.925    & 25.160  & 25.341  & 24.632   & 23.556  & 23.590   & 16.995 \\\\\n14&  J230301-305405 & 345.755  & -30.902   & 48.28    & 24.559  & 25.334  & 24.356  & 22.638  & 23.045 & 17.474 \\\\\n\n\\hline\n\\multicolumn{11}{c}{$z\\sim6$ Sample  ($i$ dropouts)} \\\\\n\\hline\n\n1& J223445-332701  & 338.687  & -33.450   & 0.418    & 25.773  &            & 24.113  & 24.442   & 22.824  & 18.457 \\\\\n2& J223719-333857  & 339.330  & -33.649   & 0.253    & 24.446   & 25.272  & 24.209 & 24.738  & 22.934  & 16.201 \\\\\n3& J224115-303408  & 340.313  & -30.569   & 1.534    &            & 25.440   & 24.481  & 24.176  & 22.531  & 17.176 \\\\\n4& J224130-302918  & 340.377 & -30.488   & 0.414    & 24.261 & 25.51  & 24.133  & 23.502  & 22.001  & 16.495 \\\\\n5& J224652-340238  & 341.717  & -34.044   & 0.364    &            & 25.277  & 24.919  & 24.649  & 22.740  & 17.127 \\\\\n6& J224957-332626  & 342.487  & -33.441   & 0.348    &            &            & 24.518  & 24.346  & 22.288  & 17.592 \\\\\n7&  J230246-293923 & 345.693  & -29.656   & 0.255    & 24.531  & 25.785  & 24.542  & 24.67  & 23.152  & 17.14  \\\\\n8&  J230423-322732 & 346.098  & -32.459    & 0.323    &            & 25.833  & 24.599  & 24.326  & 22.393  & 17.412 \\\\\n9&  J230617-335108 & 346.574  & -33.852   & 1.338    & 24.927  & 26.045    & 24.540 & 24.729  & 22.707   & 17.383 \\\\\n10&  J231051-334628 & 347.713  & -33.775   & 2.471    &            & 25.472  & 24.659  & 25.338 & 23.557 & 18.277 \\\\\n11&  J231125-342657 & 347.857  & -34.449   & 0.212    & 25.349   & 25.270   & 24.544  & 24.442  & 22.541  & 16.731 \\\\\n12&  J231245-291817 & 348.188  & -29.305   & 0.447    &            & 25.961  & 24.485  & 24.084 & 22.301  & 15.885 \\\\\n13&  J231004-334153 & 347.519  & -33.698   & 0.287    &            & 25.541  & 25.099  & 24.757 & 23.125  &        \\\\\n14& J223430-311836  & 338.627  & -31.310   & 1.23     & 25.057  & 25.216  & 24.858 & 25.361  & 23.528 &        \\\\\n15&  J232457-322915 & 351.241  & -32.488   & 0.353    & 25.209    & 25.133  & 24.645   & 24.775   & 23.102  &        \\\\\n\n\n \\hline\n \\label{tab:catalog}\n\\end{longtable}\n\\twocolumn\n\n\\begin{table*}\n    \\centering\n    \\caption{  Our final $z\\sim7$ sample  ($Z$ dropouts), showing radio properties from \\ac{ASKAP} and IR photometry from VIKING and CATWISE catalogues respectively. Missing WISE magnitudes (W1\/W2) indicate no counterpart within 2~arcsec of the VIKING position. pGalaxy indicates the probability that the source is a galaxy, obtained from the VIKING DR5 catalogue.}\n    \\begin{tabular}{cccccccccccc}\n    \\hline\n  Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5} & Mag\\_Z & Mag\\_Y & Mag\\_J & Mag\\_Ks & W1\\_mag & W2\\_mag &pGalaxy \\\\\n  & name & (deg) & (deg) & (mJy) & (AB) &(AB) & (AB)&(AB) &(Vega) & (Vega) &  \\\\\n         \\hline\n1 &  J230535--341213 & 346.399453 & -34.203674 & 0.251 & 22.53 &21.16  &21.65  &23.08 & -- &-- & $9.5\\times10^{-6}$  \\\\\n2 &  J223833--320822 & 339.63998 & -32.139507 & 0.335 & 21.41 &20.19 &19.86 &21.99 &17.09 &16.10 &0.00017  \\\\ \n3 &  J231818--311345 & 349.576502 & -31.229435 & 0.662 &21.91  &20.78 &20.79 &22.35 &18.02 &18.11 &0.000171 \\\\ \n\\hline\n    \\end{tabular}\n    \\label{tab:z7_sample}\n\\end{table*}\n\n\\section{Discussion}\n\\label{sec:discuss}\n\n\\subsection{Radio flux density}\n\\label{sec:radio_f_de}\n\n\\begin{figure}\n \\centering\n\n \\includegraphics[width=0.45\\textwidth]{flux_density_dist.png} \n \\caption{887.5\\,MHz Total radio flux density distribution of our final sample of   148 \\ac{HzRS} candidates and a known radio-quasar (selected as a Z-dropout).}\n \\label{fig:fluxD}\n\\end{figure}\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_1p3_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p9_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p7_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p4_final.png} \n\\end{subfigure}\\\\\n\\vspace{0.3cm}\n\\begin{subfigure}{.55\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p2_final.jpg} \n\\end{subfigure}\\vspace{-0.2cm}\n\\caption{ The 1.4\\,GHz radio luminosity distribution of our sample estimated for a set of spectral indices and known radio-\\ac{AGN}s (Appendix~\\ref{A}, yellow stars) as a function of redshift. Irrespective of spectral index, our sample is 1-2 orders of magnitude less luminous than known \\ac{AGN}s at the similar redshift. }\n\\label{fig:FD}\n\\end{figure*}\n\nThe observed 887.5\\,MHz flux density distribution of our final sample is shown in Figure~\\ref{fig:fluxD}. Our sample is selected from the G23 \\ac{ASKAP} radio catalogue and no additional radio properties are used in the selection.\nThe distribution of flux densities of such a flux-limited sample is expected to peak at low flux densities, and this is seen in our \\acp{HzRS} sample, which peaks at 0.2--0.4\\,mJy.\n\n Although the $z \\sim 5$ and $z \\sim 6$ sample  are smaller, they also peak at low flux densities and follow a distribution similar to that of $z\\sim4$. We  confirmed the statistical significance of this similarity by performing a Kolmogorov-Smirnov (KS) test, comparing the $z \\sim 5$ and $z \\sim 6$ samples with the $z \\sim 4$ sample. Using the KS test, the probability that the $z \\sim 5$, $z \\sim 6$, and $z\\sim7$ flux density distributions are drawn from the same population as the $z \\sim 4$ sample is 0.44, 0.31, and 0.65. Thus, the test suggests that the $z \\sim 4$, 5, 6, and 7 sources are most likely to represent a similar type of radio source population.\n\n\n \n\nWe calculate the rest-frame 1.4\\,GHz luminosity of our sources from the total flux density observed at 887.5\\,MHz, using Equation~\\ref{eq:lum_1.4GHz}, by assuming a power-law ($S_\\nu \\propto \\nu^\\alpha$) radio spectral energy distribution (SED) and adopting a  set of radio spectral indices $\\alpha =\\{-1.3, -0.9, -0.7, -0.4, 0.2 \\}$, as indicative of our complete \\acp{HzRS} sample. We assumed redshift, $z=$ 4, 5, 6, and 7 for \\textit{g, r, i,} and Z dropouts respectively in the Equation~\\ref{eq:lum_1.4GHz}. The luminosity distance (D\\textsubscript{L}) is calculated using the online cosmology calculator \\citep{wright2006cosmology}. In other words:\n\n\\begin{align}\n  L_{\\nu_{1}} &= \\frac{4\\pi D_\\textrm{L}^{2}}{(1+z)^{(1+\\alpha)}} \\left ( \\frac{\\nu_{1}}{\\nu_{2}}\\right)^{\\alpha}S_{\\nu_{2}} (\\textrm {W\/Hz}),\n  \\label{eq:lum_1.4GHz}\n\\end{align}\nwhere L\\textsubscript{$\\nu$\\textsubscript{1}} is the radio luminosity at rest-frame $\\nu\\textsubscript{1}$ derived from the observed flux density S\\textsubscript{$\\nu$\\textsubscript{2}} at $\\nu\\textsubscript{2}$ and  (1+z)\\textsuperscript{-($1+\\alpha$)} denotes the standard radio K-correction.\n\n The resulting 1.4\\,GHz luminosity of our sources for a given spectral index, shown in Figure~\\ref{fig:FD},  as a function of redshift indicates that our sample probes less powerful \\acp{HzRS} compared to known radio sources (listed in the Appendix) at the same redshift. This is expected because of the greater sensitivity of the \\ac{ASKAP} survey. Very little is known about the properties of \\acp{HzRS} with $L_{1.4} < 10^{26}$\\,W\/Hz as the known \\acp{HzRS} are mostly brighter. \n\nWe therefore used the following diagnostics to investigate the physical origin of radio emission in our sample: (i) radio luminosity at 1.4\\,GHz, (ii) 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio, (iii) WISE colour, (iv) FIR detection at 250\\,$\\mu$m, and (v) SED modelling. \n\n\n\n\n\n\n\\subsection{1.4~GHz luminosity diagnostic }\n\nIf we assume that all the radio emission is caused by star formation processes, we can calculate the \\ac{SFR} following \\citet{bell03},\n \\begin{align}\n SFR(L\\textsubscript{1.4})& = L\\textsubscript{1.4} \\times 5.52 \\times 10^{-22} M\\textsubscript{$\\odot$}\/yr.\n \\label{eq:sfr}\n\\end{align}\nWe show the results in Table~\\ref{tab:sfr}.\n\nAmong extragalactic objects, \\acp{SMG} have the highest reported \\ac{SFR}: $\\sim 6000~$M\\textsubscript{$\\odot$}\/yr \\citep{barger2014there}. By comparison, most Lyman dropouts do not have as high \\ac{SFR} as SMGs, with a maximum of $\\sim 300~$M\\textsubscript{$\\odot$}\/yr \\citep{barger2014there}. Furthermore,  \\citet[Figure~23]{barger2014there} demonstrated a turn-down in the observed \\ac{SFR} distribution function beyond 2000\\,M\\textsubscript{$\\odot$}\/yr. In Table~\\ref{tab:sfr}, we present the \\acp{SFR} estimated  for each dropout at the following two flux densities at a given spectral index: (i) the minimum of 887.5\\,MHz flux density distribution for each dropout (ii) flux density which marks the  exceeding of \\ac{SFR} beyond the highest known limit (6000\\,M\\textsubscript{$\\odot$}\/yr); \\cite{barger2014there}. In some cases, the minimum of 887.5\\,MHz flux density distribution itself results in an unphysical SFR. \nThus, most of the \\ac{SFR} shown in Table~\\ref{tab:sfr} exceed by far the highest-known \\ac{SFR} of a galaxy \\citep{barger2014there}, and we conclude that most of the radio emission from these galaxies is unlikely to be primarily generated by star formation,  implying the presence of a radio-\\ac{AGN} as well.\n\nThis finding is also consistent with the simulation \\citep{bonaldi2019tiered,wilman2008semi} of the redshift evolution of radio sources, which predicts that star forming galaxies (SFGs) are a negligible fraction of the observed radio sources beyond redshift $z > 2$ at the flux limit of 0.1\\,mJy. \n  We acknowledge that \\textit{radio}-\\ac{SFR} of the faintest end ($S_{887}\\le0.2$\\,mJy; 5 sources) of our $z\\sim4$ sample lie in the range $\\sim$4000-8000\\,M\\textsubscript{$\\odot$}\/yr,  which is very high but in priciple possible given that estimated space density of \\acp{SMG} with SFRs above 2000\\,M\\textsubscript{$\\odot$}\/yr is  $1.4\\times10^{-5}$\\,M\\textsubscript{$\\odot$}\/yr\/Mpc\\textsuperscript{3} \\citep{fu2013rapid}. Therefore, it is likely that faintest end of $z\\sim4$ sample  have a \\ac{SB} component or be pure \\acp{SB}.   \n\n \n \n\n\\begin{table}\n \\centering\n \\caption{ \n \\ac{SFR} estimated for different flux densities at $z \\sim 4$, 5, 6 and 7, assuming all radio emission is powered by star formation. \\ac{SFR} is calculated using Equations \\ref {eq:lum_1.4GHz} at spectral index, $\\alpha = -0.8$, -0.7, -0.6 given that mean spectral index of \\acp{SFG} is $\\sim0.75$ }\n \\begin{tabular}{c|c|c|c|c|c}\n \\hline\n Redshift & D\\textsubscript{L} & S\\textsubscript{887.5}& S\\textsubscript{1.4}  &L\\textsubscript{1.4} & SFR\\textsubscript{1.4}\\\\\n  $z$ & (10\\textsuperscript{3}Mpc) & (mJy) & (mJy) &  (W~Hz$^{-1}$) & (10\\textsuperscript{3}M\\textsubscript{$\\odot$}\/yr)\\\\\n  \\hline \n \n \n \n\n   \\multicolumn{6}{c}{$\\alpha=-0.8$} \\\\\n \\hline\n 4 & 36.6 & 0.15 & 0.10 &1.211$\\times 10^{25}$ & 6.67 \\\\\n\n 5 & 47.6   &0.2 &0.14 & 2.63$\\times 10^{25}$ &14.55 \\\\\n\n 6 & 58.98  & 0.2 &0.14 &3.92$\\times 10^{25}$  &21.62 \\\\\n \n7 & 70.54   & 0.2 &0.14 &5.45$\\times 10^{25}$  &30.11 \\\\\n\n  \\hline\n  \n   \\multicolumn{6}{c}{$\\alpha=-0.7$} \\\\\n \\hline\n  \\multirow{2}{*}{4} & \\multirow{2}{*}{36.6} & 0.15 & 0.11 &1.07$\\times 10^{25}$ & 5.95 \\\\\n       &       & 0.2 & 0.15  &1.44$\\times 10^{25}$ & 7.93\\\\\n\\hline\n  5 & 47.6  &0.2 &0.15 & 2.31$\\times 10^{25}$ &12.73 \\\\\n\n  6 & 58.98  & 0.2 &0.15 &3.37$\\times 10^{25}$  &18.63 \\\\\n \n7 & 70.54  & 0.2 &0.15 &4.64$\\times 10^{25}$  &25.53 \\\\\n  \\hline\n  \n   \\multicolumn{6}{c}{$\\alpha=-0.6$} \\\\\n \\hline\n  \\multirow{2}{*}{4} & \\multirow{2}{*}{36.6} & 0.15 & 0.11 &6.9$\\times 10^{24}$ & 3.84 \\\\\n       &       & 0.24 & 0.18  &1.11$\\times 10^{25}$ & 6.15\\\\\n\\hline\n\n 5 & 47.6   &0.2 &0.15 & 1.41$\\times 10^{25}$ &7.78 \\\\\n\n 6 & 58.98  & 0.2 &0.15 &1.97$\\times 10^{25}$  &10.87 \\\\\n \n7 & 70.54   & 0.2 &0.15 &2.6$\\times 10^{25}$  &14.36 \\\\\n\n  \\hline\n \\end{tabular}\n \\label{tab:sfr}\n\\end{table}\n\n\n\n\\subsection{Radio-to-3.4\\,$\\mu$m flux density ratio}\n\\label{sec:NIR}\n\n\n\\citet[][Figure 4]{norris2011deep} presented a new measure for radio-loudness of a source, 1.4\\,GHz-to-3.6\\,$\\mu$m flux density ratios. They compared the 1.4\\,GHz-to-3.6\\,$\\mu$m flux density ratios for different galaxy populations  namely, \\acp{HzRG}, \\acp{SMG}, \\acp{SB}, RL \\& RQ quasars and \\acp{ULIRG}, as a function of redshift.  They showed that galaxies powered primarily by star forming activity have a lower ratio in contrast to radio-AGNs at all redshifts. \n We use 3.4\\,$\\mu$m photometry from CATWISE2020 catalogue \\citep{marocco2021catwise2020} since there is no substantial difference between 3.4 \\& 3.6\\,$\\mu$m passbands, and 3.4\\,$\\mu$m magnitude (W1) is converted to flux density \\citep{cutri2012explanatory} using,\n\\begin{align}\n  S\\textsubscript{3.4\\,$\\mu$m} = 306.682 \\times 10^{-W1\/2.5} ~ \\textrm{Jy.}\n\\end{align}.\n\n\nWe compare \\cite{norris2011deep} model with the 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio of our sample at redshifts, $ z \\sim 4$, 5, and 6 in Figure~\\ref{fig:radio_ir_ratio}.  The measured flux density at 887.5\\,MHz was used to estimate the 1.4\\,GHz flux density, assuming spectral indices,  $\\alpha$ of -1.3, -0.7, and 0.2 as indicative of our entire \\ac{HzRS} sample.\nAccording to that model, radio-\\acp{AGN} have a radio-to-3.6\\,$\\mu$m flux density ratio above 100 at all redshifts. However, \\cite{maini2016infrared} showed that the ratio for Type 1 \\& 2 \\ac{AGN}\/QSO and high power radio galaxies start to decrease (between 10 and 100) beyond $z\\sim4$ and $z\\sim5$ . We contend that reason for this difference is that the \\cite{norris2011deep} model refers to \\emph{powerful} radio-\\ac{AGN}s only, at all redshifts.  About $55\\%$ of our sample have a 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio between 1 and 10 and $\\sim41\\%$ between 10 and 100. Only 5 sources in our sample have a radio-to-3.4\\,$\\mu$m ratio greater than 100. \n\n\nIt therefore seems that high-$z$ radio-\\ac{AGN}s can extend to even lower values ($<10$) of radio\/IR ratios than previously thought.  The following are some possible explanations for their low ratio: (i) radio-\\ac{AGN} is accompanied by SB activity, and the optical emission being redshifted to W1 band causing an increase in the 3.4\\,$\\mu$m flux density (ii) the inherently low radio powers of high-$z$ radio-\\acp{AGN}, as predicted by simulation in \\cite{saxena2017modelling} (iii) optical emission from un-obscured quasars at $z\\ge4$ being redshifted to W1 band. We discuss this further in Section \\ref{sec:FIR}.\n\n\n \\begin{figure*}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_1p3.png} \n\\end{subfigure}\n\\hspace{0.1cm}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_0p7.png} \n\\end{subfigure}\n\\hspace{0.1cm}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_0p2.png} \n\\end{subfigure}\n\n \\caption{ 887.5\\,MHz flux density vs. 3.4\\,$\\mu$m flux density for $z\\gtrsim4$ sample, colour coded according to their 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio. 1.4\\,GHz flux density is estimated from observed flux density at 887.5\\,MHz assuming spectral indices, $\\alpha = -1.3$, -- 0.7, 0.2.  }\n \\label{fig:radio_ir_ratio}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\\subsection{FIR detection}\nDust heated by stars or \\ac{AGN} generally re-radiates the absorbed energy at FIR wavelengths. Each galaxy will have a distribution of dust temperatures which is determined by the size and distribution of dust with respect to the heating source (\\ac{SF} or \\ac{AGN}). Cool ($T\\sim20\\,K$) dust arises from the diffuse interstellar medium (ISM), warm ($T\\sim40\\,K$) dust from SF regions, and even warmer $T\\sim60-100\\,K$ dust results from \\ac{AGN} activity. \n\nWe queried the {\\it Herschel} database to check whether any of our sources have been detected at FIR wavelengths. We searched the SPIRE point source catalogues at 250\\,$\\mu$m, 350\\,$\\mu$m, and 500\\,$\\mu$m \\citep{schulz2017spire} separately in a 5~arcsec cone around the WISE position of each of our HzRS sample,  which yielded a false-id rate of $\\sim16\\%$. Several of our sources have a SPIRE counterpart, but none was found in the PACS point source catalogues.  A total of 14 and 3 sources from our $z\\sim4$ and $z\\sim6$ sample respectively were found to have a SPIRE counterpart, representing only 10\\% of our entire \\ac{HzRS} sample.\n \nWe present the results in Table~\\ref{tab:spire_sources}.  The SPIRE 250\\,$\\mu$m band probes a rest wavelength $\\sim$45 - 63\\,$\\mu$m  at $z=3-4.3$ ($g$ dropouts) and $\\sim$34 - 38\\,$\\mu$m  at $z=5.6-6.2$ ($i$ dropouts). At wavelength $<50$\\,$\\mu$m in the rest-frame, an \\ac{AGN} heated dust torus \nmay contribute significantly to the FIR emission.\nTherefore the SPIRE detection of our radio sources suggests that the observed 250\\,$\\mu$m emission may arise from either (i) a dust torus heated by an \\ac{AGN} or (ii) \\ac{SB} activity or (iii) a combination of both \\ac{SF} and \\ac{AGN}.\n\n\\begin{table}\n    \\centering\n     \\caption{Our \\acp{HzRS} with a FIR detection in either of {\\it Herschel}-SPIRE 250\\,$\\mu$m, 350\\,$\\mu$m $\\&$ 500\\,$\\mu$m bands, selected from a 5~arcsec search around their WISE position at which false-ID rate is $\\sim16\\%$.  The probability of a source's measured SPIRE flux being contaminated by a nearby neighbour is about 1\/14.}\n    \\begin{tabular}{c|c|c|c}\n    \\hline\n        Source ID & S\\textsubscript{250} & S\\textsubscript{350} & S\\textsubscript{500}  \\\\\n         &  (mJy) & (mJy) & (mJy)   \\\\\n        \n        \\hline\n\\multicolumn{4}{c}{$z\\sim4$ candidates}        \\\\\n\\hline\n\nJ224308-313622 & 59.3 &\t &  \\\\\nJ223533-343305 & 67.4 & 40.7 &  --    \\\\\nJ231210-332436 & 49.9 &\t32.3 &   --   \\\\\nJ224203-333606 & 59.5 &\t68.5 &  --    \\\\\nJ231826-323746 & 39.9 & -- & -- \\\\\nJ223729-325838 & 34.8 & -- & -- \\\\\nJ224017-325128 & 66.2 & -- & -- \\\\\nJ224552-312733 & 51.2 &--   &--   \\\\\nJ225343-313305 & 65.4     & --  & --  \\\\\nJ231648-303629 & 65.6     & --  &--   \\\\\nJ232019-294205 & 93.6     &77.2   &--   \\\\\nJ232135-320117 & 44.5     & --   &   \\\\\nJ225915-314116 & --& 78.1 & 62.9 \\\\\nJ224955-294536 & -- & -- & 49 \\\\\n\n\\hline\n\\multicolumn{4}{c}{$z\\sim6$ candidates}  \\\\\n\\hline \nJ224652-340238 & 92.9 & 62.4 & 47.9 \\\\\nJ231245-291817 & 98.5 & 86.8 & 48.9 \\\\\nJ223719-333857 & 65.6 & 52.5 & -- \\\\\n\\hline\n    \\end{tabular}\n    \\label{tab:spire_sources}\n\\end{table}\n\n\\subsubsection{Radio-FIR relation}\n\n\nIn this section, we explore the 250\\,$\\mu$m-to-1.4\\,GHz luminosity ratio of our sources, as defined by Equation~\\ref{eq:q250},  which is a tracer of star formation activity. Here we are only interested in order-of-magnitude estimates and hence we neglect evolution. Another paper in which we model evolutionary effects is in preparation. \n$q\\textsubscript{250}$ is defined as:\n\\begin{align}\n  q_{250} = \\log_{10}\\left (\\frac{L_{250}}{L_{1.4}}\\right ) ~,\n  \\label{eq:q250}\n\\end{align}\nwhere $L\\textsubscript{250}$ and $L\\textsubscript{1.4}$ are the rest-frame luminosities at 250\\,$\\mu$n and 1.4\\,GHz respectively.\n\nTo estimate rest-frame $L\\textsubscript{250}$, we must first estimate the \nK-correction (k\\textsubscript{corr}) defined as \n\\begin{align}\n  k_{corr} = \\frac{S_{rest}}{S_{obs}} ~,\n  \\label{eq:k_corr}\n\\end{align}\nwhere $S_{\\rm obs}$ and $S_{\\rm rest}$ are the observed and the rest-frame flux densities respectively. \n\nAssuming a greybody thermal emission, the spectral flux density at a given frequency and temperature, $S_{\\nu}(T)$, would be a modified Planck's radiation law $B(\\nu,T)$,\n\\begin{align}\n S_{\\nu}(T) &\\propto \\nu^{\\beta}B(\\nu,T) ~, \\label{eq:S_nu}\\\\  \\intertext{where}\n B(\\nu,T) &= \\frac{2h}{c^{2}}\\frac{\\nu^{3}}{e^{\\frac{h\\nu}{k_{B}T}}-1}\n \\label{eq:planck}\n\\end{align}\n and $\\beta$ represents the dust emissivity index, for which we assume a value of  1.5 \\citep{kirkpatrick2015role}.\n\n\nWe calculate the K-correction using Equations~\\ref{eq:k_corr}, \\ref{eq:S_nu} and \\ref{eq:planck}, assuming 2 dust temperatures, 45\\, K, $\\&$ 80\\,K, corresponding to warm and warmer dust components  based on \\cite{kirkpatrick2015role} study, and present the results in Table~\\ref{tab:k_corr}.\n\\begin{table}\n \\centering\n \\caption{K-correction estimated for the observed frame at 250\\,$\\mu$m,  assuming the dust emission either dominated by a warmer dust ($T_{w} \\sim$ 80\\,K) or a warm dust ($T_{w} \\sim$ 45\\,K).}\n\n \\begin{tabular}{c|c|c|c|c}\n \\hline\n  redshift & $\\nu$\\textsubscript{obs} & $\\lambda$\\textsubscript{rest} & $\\nu$\\textsubscript{rest}  & k\\textsubscript{corr} \\\\\n  ($z$) & (Hz) & ($\\mu$m) & (Hz)  & \\\\\n\\hline\n  \\multicolumn{5}{c}{$T_{dust} \\sim 45\\,K$} \\\\\n  \\hline\n   4 &\\multirow{3}{*}{$1.2\\times10^{12}$} &50 & $5.9\\times10^{12}$ &  6.22 \\\\\n  5 & & 42 & $7.2\\times10^{12}$  & 3.8 \\\\\n  6 & &36 & $8.4\\times10^{12}$  &  2.11\\\\\n  \\hline\n  \\multicolumn{5}{c}{$T_{dust} \\sim 80\\,K$} \\\\\n  \\hline\n  4 &\\multirow{3}{*}{$1.2\\times10^{12}$} &50 & $5.9\\times10^{12}$ &  40.77 \\\\\n  5 & & 42 & $7.2\\times10^{12}$  & 44.93 \\\\\n  6 & &36 & $8.4\\times10^{12}$  & 43.46 \\\\\n  \\hline\n \\end{tabular}\n \\label{tab:k_corr}\n\\end{table}\n\nWe then calculate the luminosity $L_{250}$,\n\\begin{align}\n  L_{250} = \\frac {4\\pi S_{250}K_{cor} D_\\textrm{L}^{2}}{1+z} ~, \n  \\label{eq:L250}\n\\end{align}\nwhere $S_{250}$ is the observed flux densities of our sources in the SPIRE database and D\\textsubscript{L} is the luminosity distance. \n\n\n\n The resulting $q\\textsubscript{250}$ estimated for two extremes of assumed radio spectral indices (see Section~\\ref{sec:radio_f_de}) is shown in Figure~\\ref{fig:q250}. All sources with a reliable detection as indicated by SNR>5 and 3>SNR>5 have q$_{250}$ exceeding the threshold of 1.3 \\citep{jarvis2010herschel, virdee2013herschel} indicating luminous SFG.   The high values of $q\\textsubscript{250}$,  irrespective of radio spectral index, suggest the presence of an active \\ac{SB} component in SPIRE detected sources but on the other hand our K-correction is very uncertain because of our lack of knowledge of the FIR SED, and so we do not consider this a definitive indicator of SF. Furthermore, SPIRE detected sources have $S_{887.5} > 0.2$\\,mJy implying the presence of a radio-\\ac{AGN} component according to Table~\\ref{tab:sfr}.   Therefore, we attempt to estimate \\ac{SFR} (see Section~\\ref{sec:FIR}), from the observed 250\\,$\\mu$m emission to verify whether the resulting SFR is sufficient enough to generate the observed radio luminosity. We also note  that q\\textsubscript{250} criterion is based on the study of powerful radio-\\acp{AGN} in the literature, whereas this study probe low power radio-\\acp{AGN}, suggesting q$_{250}>1.3$ may not be applicable in this case.  \n\n\n\n \\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_45k_1p3_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_45k_0p2_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_80k_1p3_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_80k_0p2_final.png} \n\\end{subfigure}\n \\caption{ 887.5\\,MHz flux density vs. 250\\,$\\mu$m flux density for $z\\gtrsim4$ sample, colour coded according to their q\\textsubscript{250} measurements.  }\n \\label{fig:q250}\n\\end{figure*}\n\n\\subsubsection{\\ac{SFR} from S\\textsubscript{250}}\n\\label{sec:FIR}\n\nWe estimate the SFR from S\\textsubscript{250} via the rest-frame 24\\,$\\mu$m luminosity, $L_{24\\,\\mu m}$ (in erg\/s), using the following equations from \\citet{brown2017calibration},\n\\begin{align}\n \\log L_{24_{\\mu m}} &= 40.93+1.3(\\log L_{H\\alpha} - 40)\n \\label{eq:Halpha}\\\\\n SFR &= L_{H\\alpha}\\times5.5\\times10^{-42}~\\rm{M_\\odot \/yr}~.\n  \\label{eq:sfr_ha}\n\\end{align}\n\nFor the calculation of $\\text{L}_{24\\,\\mu m}$, we consider 4 different spectral energy distribution (SED) models corresponding to: (i) an ultra luminous infrared galaxy (Arp~220), (ii) a starburst (M\\,82),  (iii) Mrk\\,231 (luminous infrared \\ac{AGN}), and (iv) a composite system (IRAS F00183--7111), in which the optical\/IR SED is dominated by the starburst surrounding the \\ac{AGN}, with the \\ac{AGN} emerging only at radio wavelengths \\citep{norris12}. In each case, we interpolate the SED using flux density measurements obtained via NASA\/IPAC Extragalactic Database (NED). \n\n\nWe perform the following calculations to estimate $\\text{L}_{24\\,\\mu m}$ of our sources:\n\n\\begin{enumerate}\n \\item We calculate the luminosity of our sources corresponding to their emitted wavelengths from their observed flux density at 250\\,$\\mu$m using the equation,\n \n \\( L_{\\lambda} = \\frac{4\\pi D_\\textrm{L}^{2}S_{250}}{1+z}\\), \\\\\n where $\\lambda$ represents the respective emitted (or rest-frame) wavelength at $z\\sim4,5,6$ as shown in column 3 of  Table~\\ref{tab:k_corr}.\n \\item For each model source (Arp~220, M\\,82, Mrk\\,231, IRAS F00183--7111) we assume that the ratio $L_{24\\mu m}\/L_{\\lambda}$ for our sources is the same as that of the model source as measured from their SED: \n \n \n  \\(\\left(\\frac{L_{24\\mu m}}{L_{\\lambda}}\\right)_{our source}\\) =\n  \\(\\left(\\frac{S_{24\\mu m}}{S_{\\lambda}}\\right)_{model}\\).\n \n\\end{enumerate}\n\n\n \n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[trim={15 210 15 0},width=\\textwidth]{SFR_z4z6.pdf}\n    \\caption{ \\ac{SFR} estimated from S\\textsubscript{250} by assuming various SED models.}\n    \\label{fig:sfr}\n\\end{figure*}\nWe present the results of this calculation in Figure~\\ref{fig:sfr}. The resulting \\ac{SFR}s  for $z\\sim4$ \\& $z\\sim6$ samples lie in the range $\\sim 400-3500$\\, M\\textsubscript{$\\odot$}\/yr \\& $\\sim 2000-8200$\\, M\\textsubscript{$\\odot$}\/yr respectively, which is high but not unphysical, and similar to the \\ac{SFR}s reported for \\ac{SB}s and SMGs at similar redshifts \\citep{barger2014there}. These SFRs would generate a radio power in the order of  $\\sim 10^{24}-10^{25}$~W~Hz$^{-1}$, which is typically a small fraction ($\\le 10\\%$) of the observed radio luminosities.\n\n We conclude that these galaxies detected by \\textit{Herschel}-SPIRE are likely to be composite galaxies containing both a radio-\\ac{AGN} and a starburst. We further note that the radio-to-3.4\\,$\\mu$m flux density ratio of SPIRE detected sources is $<100$. 7 out of 15 250\\,$\\mu$m detected sources found to have $W1-W2 > 0.8$ WISE colour, suggesting an \\ac{AGN} component. At $z\\gtrsim4$, W1 WISE band observations probe the optical rest-frame including H$\\alpha$ emission, an indicator of SF activity or  quasar light or both. This suggests that a fraction of  high-$z$ radio-\\ac{AGN}s ($\\log L\\textsubscript{1.4}=25$) at $z\\gtrsim4$ may have a \\ac{SB} host, contrary to low-z radio-\\ac{AGN}s which are usually hosted by quiescent galaxies. Similarly, \\cite{rees2016radio} \nfound that $z > 1.5$ radio-\\ac{AGN}s tend to be hosted by SFGs. Our study suggests that this may be true at even higher redshifts.\n\n\n \n\\subsection{WISE colours}\n\\label{sec:wise_colours}\n\n\n\nMid-IR colour selection criteria are an efficient tool to identify a hot accretion disk, which is an indicator of \\ac{AGN} in galaxies. The dust reprocesses the emission from the accretion disk into the IR, which dominates the \\ac{AGN} SED at wavelengths from $\\sim1$\\,$\\mu$m to a few tens of microns, the wavelength regime covered by the WISE survey. \n\n\\cite{stern2012mid} demonstrated that a simple WISE colour cut, $W1-W2>0.8$, selects \\ac{AGN} with a reliability of 95\\% and with a completeness of 78\\%. However, this colour cut works efficiently for low-z sources ($z\\sim1$) only, given that is defined using observed fluxes not rest-frame ones. \\cite{stern2012mid} therefore discussed a number of possible scenarios to interpret W1-W2 colour at higher redshifts.\n\nFollowing are the possibilities, taken from \\cite{stern2012mid}, applicable to our $z\\gtrsim4$ sample, \n\\begin{enumerate}\n \\item At $z\\gtrsim3.5$, W1\/W2 bands probe optical \\& near-IR rest-frame emission\n  causing the $\\sim1$\\,$\\mu$m minimum  seen in some starburst galaxies\n shifting to W2 band and H$\\alpha$ emission shifting to W1. This results in a blue W1-W2 (<0.8) colour \n\n \\item A highly obscured \\ac{AGN} will have $W1-W2>0.8$ at all redshifts.\n \\item For a composite system (\\ac{AGN} $+$ \\ac{SB}), dilution by host-galaxy light can limit the identification of low luminosity AGNs.\n\\end{enumerate}\n \n\nWe looked at the W1-W2 colour of our sources, utilizing CATWISE magnitudes. We considered only those sources without any flags and with SNR$\\ge$5 in the W1 and W2 bands. This identifies 106 sources, of which only 28 sources satisfy the $W1-W2>0.8$ \\ac{AGN} criterion. We further found that all but 2 of the sources satisfying the $W1-W2>0.8$ \\ac{AGN} criterion have S\\textsubscript{887}$\\gtrsim$0.2-2.5\\,mJy and  1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio between 1 and 100. This supports our argument \nthat (i) low power radio sources in our sample are powered by a radio-\\ac{AGN} (ii) the radio-to-IR flux density ratio of low power radio-\\acp{AGN} can extend to lower values (<100). On the other hand, the sources with $W1-W2<0.8$ could be either unobscured quasars, as they have a blue $W1-W2$ colour at high-$z$ as demonstrated in \\cite{ross2020near} or the composite galaxies where the SB component results in a blue $W1-W2$ colour.\nIn addition to this, we verified the WISE magnitudes of a known starbust at $z=4.1$ \\citep{ciesla2020hyper} and found that its $W1-W2 = 0.63$ matches our $z\\gtrsim4$ sample with S\\textsubscript{1.4}$<$1\\,mJy.\n\n\n\n\n\n\n\n\n\n\\subsection{SED modelling}\n\\label{sec:sed_modelling}\n\n It is beyond the scope of this paper to perform full SED modelling of these galaxies, but here we make an illustrative comparison of these galaxies with two representative low-redshift galaxies. To do so, we shift two local radio sources to $z\\gtrsim4$ for comparison: (i) Cygnus~A, a radio galaxy at $z = 0.05607$, and (ii) IRAS F00183--7111, an ultraluminous infrared galaxy (ULIRG) at $z=0.327$ showing composite emission from a radio loud-\\ac{AGN} and a \\ac{SB}. Each of these sources is distinct: their radio emission either comes from an \\ac{AGN} or a combination of SF and \\ac{AGN}. Their broadband observed photometry is obtained from NED.  We added more radio data from  \\citet[Table~1]{norris2012radio} to the IRAS F00183--7111 template since the radio coverage is poor and this wavelength regime is critical for this analysis. \n\nWe do not consider galaxies exclusively powered by star formation activity, like M\\,82 and Arp~220, as they produce radio flux densities three or four orders of magnitude below the current detection limit when shifted to $z\\gtrsim4$, indicating that none of our sources represent typical SFGs. \n \nWe create the SED at $z\\gtrsim4$ by shifting the observed SED in the frequency space by a factor of \\textit{(1+$z$)\\textsuperscript{-1}}, and in flux density space by a factor of (\\textit{D\\textsubscript{obs}}\/\\textit{D\\textsubscript{z})\\textsuperscript{2}}$\\times {\\it (1+z_{\\rm {\\it shift}}}\/{\\it 1+z_{\\rm {\\it obs}}})$.\\textit{ D\\textsubscript{obs}} and \\textit{D\\textsubscript{z}} are the luminosity distance to the observed redshift and shifted redshift ($z_{\\rm {\\it shift}}\\sim4$, 5, 6 or 7) respectively, estimated using an online cosmology calculator \\citep{wright2006cosmology}.\n\n\nFigure~\\ref{fig:shifted_SED} shows the resulting shifted SEDs. The extracted 1.4\\,GHz flux density from the shifted SEDs, shown in Table~\\ref{tab:shifted_sed}, demonstrates that (i) active galaxies powered by a radio-\\ac{AGN}, such as Cygnus~A, can represent the brighter sources in our sample and (ii) a composite system like IRAS F00183-7111 consisting of a radio-\\ac{AGN} and a significant \\ac{SB} component, can represent fainter sources in our sample. Thus, our SED modelling shows that a radio-\\ac{AGN} and a composite system can reproduce the galaxies in our sample implying that properties of our \\acp{HzRS}  are not that extraordinary compared to typical local galaxies. At the same time, we note that IRAS F00183-7111 does not reproduce our sample's observed SPIRE flux densities, as its SFR ($\\sim 220$\\,$M_{\\odot}$\/yr; \\cite{mao2014star}) is not sufficient.\n\n\n\n\\begin{figure*}\n \\centering\n  \\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{CygnusA_4_final.png} \n\\end{subfigure}\n\\hspace{0.5cm}\n \\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{f00183_3_final.png} \n\\end{subfigure}\n \\caption{Template SEDs shifted to redshifts, $z=4, 5, 6, 7$ in grey, black, red and blue colours respectively. SED templates are built by simply connecting the datapoints and performed 1D interpolation using python-scipy (interp1d) package to obtain 1.4\\,GHz flux density information from the shifted SEDs. 1.4\\,GHz is marked in black solid line. Pink points represent the observed SED, obtained from NED.\n }\n \\label{fig:shifted_SED}\n\\end{figure*}\n\\begin{table}\n \\centering\n  \\caption{1.4\\,GHz flux density extracted from the template SEDs shifted to $z=4,5,6$ by performing 1d interpolation using interp1d function in python-scipy package.}\n \\begin{tabular}{c|c|c|c|c}\n \\hline\n  \\multirow{2}{*}{Source} & \\multicolumn{3}{c}{S\\textsubscript{1.4} (mJy)}\\\\\n  \\cline{2-5}\n   & $z\\sim4$ & $z\\sim5$ & $z\\sim6$ & $z\\sim7$ \\\\\n   \\hline\n  Cygnus A & 60 & 29 & 20 & 14 \\\\ \n  IRAS F00183 & 0.6 & 0.3 & 0.2 &  0.1 \\\\\n   \\hline\n \\end{tabular}\n \\label{tab:shifted_sed}\n\\end{table}\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have used the Lyman dropout technique to identify a sample of 149 radio sources at $z\\gtrsim4-7$, of which one is  a known radio-quasar (VIK J2318\u22123113; selected as a Z-dropout) at $z=6.44$, and 148 are newly-identified. Our study reveals a new population of high-redshift ($z\\sim$ 4--7) radio-\\ac{AGN}s, $\\sim1-2$ orders of magnitudes fainter than currently known radio-\\ac{AGN}s at similar redshifts, but with 1.4\\,GHz luminosities typical of lower-redshift radio-loud \\ac{AGN}. \n\nComparison with spectroscopic redshifts in the COSMOS field indicate that our $z\\sim4$  sample is about 88\\% reliable, the  $z\\sim5$  sample is about 81\\% reliable, and the  $z\\sim6$  sample is  at least about 62\\% reliable. We do not have a reliability estimate for the  $z\\sim7$  sample. \n\nWe have explored radio and IR observations to understand  the origin of the radio emission of our  \\ac{HzRS} sample.  Our conclusions are as follows.\n\n\\begin{enumerate}\n \\item The radio (1.4\\,GHz) estimated \\ac{SFR} of our sample (Table~\\ref{tab:sfr}) gives unphysical values, indicating that radio emission in our sources is not solely powered by star formation. \n \n \\item The faint ($\\log L\\textsubscript{1.4}=25$) and bright ($\\log L\\textsubscript{1.4}\\ge26$)\n end of our sample spans the low and high power radio galaxy luminosity classes respectively, suggesting the presence of a radio-\\ac{AGN} component in our sample. This study presents a new population of less powerful radio-\\ac{AGN}s candidates at $z\\sim4$, 5 and 6 that have been missed by previous surveys.\n \n\n\n \\item  $\\sim10\\%$ of our \\ac{HzRS} sample are detected in the {\\it Herschel}-SPIRE bands which probe \\ac{SB} heated dust emission and \\ac{AGN} heated dust torus emission at $z>2$. This suggests that SPIRE detected sources are likely to represent composite systems. \n \n \\item We demonstrate that some high-$z$ radio-\\ac{AGN}s tend to have hosts that are \\ac{SB} galaxies, in contrast to low-$z$ radio-\\ac{AGN}s, which are usually hosted by quiescent elliptical galaxies.\n \n  \\item  Using the $W1-W2>0.8$ AGN indicator, we identified 28 radio-\\acp{AGN},  26 of which are found to be on the faint end of the observed 887.5\\,MHz flux density distribution.  We further demonstrate that the 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio of these weak radio-\\ac{AGN}s extends to lower values (1-100) than previously thought.\n \n \\item SED modelling confirms that a composite system (radio-\\ac{AGN} $+$ \\ac{SB}) and a radio galaxy at $z\\gtrsim4$ can produce the radio flux densities similar to ones observed at the faint and bright end respectively.\n\n\\end{enumerate}\n\\section{Future work}\n\n Spectroscopic follow-up of our sample is essential (i) to confirm the redshifts of these candidate sources identified by the Lyman dropout technique (ii) to verify the reliability of magnitude cut-off introduced in $Z$-band for $i$ dropouts (iii) to establish the criteria to identify interlopers, if present. The above three goals are critical in producing reliable criteria to select \\acp{HzRS} in the full \\ac{EMU} survey and thus verifying the model \\citep{raccanelli2012cosmological} for the redshift distribution of radio sources.\n\n\n\n\\section*{Acknowledgements}\n\nThe Australian SKA Pathfinder is part of the Australia Telescope National Facility which is managed by \\ac{CSIRO}. Operation of \\ac{ASKAP} is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. \\ac{ASKAP} uses the resources of the Pawsey Supercomputing Centre. \nEstablishment of \\ac{ASKAP}, the Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. \nWe acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. \n\nBased on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples).\n\nIsabella Prandoni acknowledges support from INAF under the PRIN MAIN stream \"SAuROS\" project, and from CSIRO under its Distinguished Research Visitor Programme. \n\n We thank an anonymous referee for helpful comments and some excellent suggestions.\n\\section*{Data Availability}\n\n \n\nThis study utilised the data available in the following public domains: \n\\begin{enumerate}\n    \\item https:\/\/data.csiro.au\/domain\/casdaObservation\n    \\item https:\/\/kids.strw.leidenuniv.nl\/DR4\/access.php\n    \\item http:\/\/horus.roe.ac.uk\/vsa\/index.html\n    \\item https:\/\/irsa.ipac.caltech.edu\/\n\\end{enumerate}\n\nThe derived final datasets are available in the article.\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nIf $X_1,X_2,\\ldots,X_n \\stackrel{\\mathrm{ind}}{\\sim}\nf$ are a sample from a nonincreasing density $f$ on $[0,\\infty)$,\nthen the Grenander estimator, the nonparametric maximum likelihood\nestimator (NPMLE) $\\tilde f_n$ of $f$ [obtained by maximizing the\nlikelihood $\\prod_{i=1}^n f(X_i)$ over all nonincreasing densities],\nmay be described as follows: let $\\mathbb{F}_n$ denote the empirical\ndistribution function (EDF) of the data, and $\\tilde{F}_n$ its least\nconcave majorant. Then the NPMLE $\\tilde{f}_n$ is the left-hand\nderivative of $\\tilde{F}_n$. This result is due to Grenander (\\citeyear\n{grenander56}) and\nis described in detail by Robertson, Wright and Dykstra (\\citeyear\n{RWD88}), pages\n326--328. Prakasa Rao (\\citeyear{prakasa69}) obtained the asymptotic\ndistribution of\n$\\tilde{f}_n$, properly normalized: let $\\mathbb{W}$ be a two-sided\nstandard Brownian motion on $\\mathbb{R}$ with $\\mathbb{W}(0) = 0$ and\n\\[\n{\\mathbb C} = \\mathop{\\arg\\max}_{s\\in{\\mathbb R}} [{\\mathbb W}(s) - s^2].\n\\]\nIf $0 < t_0 < \\infty$ and $f'(t_0) \\ne0$, then\n\\begin{equation}\n\\label{eq:chrnff}\nn^{1\/3} \\{ \\tilde f_n(t_0) - f(t_0) \\} \\Rightarrow\n2 \\bigl|\\tfrac{1}{2} f(t_0) f'(t_0) \\bigr|^{1\/3} {\\mathbb C},\n\\end{equation}\nwhere $\\Rightarrow$ denotes convergence in distribution. There are\nother estimators that exhibit similar asymptotic\nproperties; for example, Chernoff's (\\citeyear{chernoff64}) estimator\nof the mode, the\nmonotone regression estimator [Brunk (\\citeyear{brunk70})],\nRousseeuw's (\\citeyear{rousseeuw84}) least median of squares estimator,\nand the estimator\nof the shorth [Andrews et al. (\\citeyear{andrewsetal72}) and Shorack\nand Wellner (\\citeyear{shorackWe86})].\nThe seminal paper by Kim and Pollard (\\citeyear{kimPo90}) unifies\n$n^{1\/3}$-rate\nof convergence problems in the more general \\mbox{$M$-estimation} framework.\nTables and a survey of statistical problems in which the distribution\nof ${\\mathbb C}$ arises are provided by Groeneboom and Wellner\n(\\citeyear{GW01}).\n\nThe presence of nuisance parameters in the limiting distribution (\\ref\n{eq:chrnff}) complicates the construction of\nconfidence intervals. Bootstrap intervals avoid the problem of\nestimating nuisance parameters and are generally reliable in problems\nwith $\\sqrt{n}$ convergence rates. See Bickel and Freedman (\\citeyear{BF81}),\nSingh (\\citeyear{singh81}), Shao and Tu (\\citeyear{shaoTu95}) and its\nreferences. Our aim in this\npaper is to study the consistency of bootstrap methods for the\nGrenander estimator with the hope that the monotone density estimation\nproblem will shed light on the behavior of bootstrap methods in similar\ncube-root convergence problems.\n\nThere has been considerable recent interest in this question. Kosorok\n(\\citeyear{kosorok07}) show that bootstrapping from the EDF ${\\mathbb\nF}_n$ does not\nlead to a consistent estimator of the distribution of $n^{1\/3} \\{\n\\tilde f_n(t_0) - f(t_0)\\}$. Lee and Pun (\\citeyear{leePun06}) explore\n$m$ out of $n$\nbootstrapping from the empirical distribution function in similar\nnonstandard problems and prove the consistency of the method. L\\'eger\nand MacGibbon (\\citeyear{legerMa06}) describe conditions for a\nresampling procedure to\nbe consistent under cube root asymptotics and assert that these\nconditions are generally not met while bootstrapping from the EDF. They\nalso propose a smoothed version of the bootstrap and show its\nconsistency for Chernoff's estimator of the mode. Abrevaya and Huang\n(\\citeyear{AH05}) show that bootstrapping from the EDF leads to inconsistent\nestimators in the setup of Kim and Pollard (\\citeyear{kimPo90}) and propose\ncorrections. Politis, Romano and Wolf (\\citeyear{PRW99}) show that subsampling\nbased confidence intervals are consistent in this scenario.\n\nOur work goes beyond that cited above as follows: we show that\nbootstrapping from the NPMLE $\\tilde{F}_n$ also leads to inconsistent\nestimators, a result that we found more surprising, since $\\tilde{F}_n$\nhas a density. Moreover, we find that \\textit{the bootstrap estimator,\nconstructed from either the EDF or NPMLE, has no limit in probability}.\nThe finding is less than a mathematical proof, because one step in the\nargument relies on simulation; but the simulations make our point\nclearly. As described in Section~\\ref{discussion}, our findings are\ninconsistent with some claims of Abrevaya and Huang (\\citeyear{AH05}).\nAlso, our\nway of tackling the main issues differs from that of the existing\nliterature: we consider conditional distributions in more detail than\nKosorok (\\citeyear{kosorok07}), who deduced inconsistency from\nproperties of\nunconditional distributions; we directly appeal to the characterization\nof the estimators and use a continuous mapping principle to deduce the\nlimiting distributions instead of using the ``switching'' argument [see\nGroeneboom (\\citeyear{groene85})] employed by Kosorok (\\citeyear\n{kosorok07}) and Abrevaya and Huang\n(\\citeyear{AH05}); and at a more technical level, we use the Hungarian\nRepresentation theorem whereas most of the other authors use empirical\nprocess techniques similar to those described by van der Vaart and\nWellner (\\citeyear{VW00}).\n\nSection \\ref{prelim} contains a uniform version of (\\ref{eq:chrnff})\nthat is used later on to study the consistency of different bootstrap\nmethods and may be of independent interest. The main results on\ninconsistency are presented in Section \\ref{boots_prob}. Sufficient\nconditions for the consistency of a bootstrap method are presented in\nSection \\ref{smooth_boots} and applied to show that bootstrapping from\nsmoothed versions of $\\tilde{F}_n$ does produce consistent estimators.\nThe $m$ out of $n$ bootstrapping procedure is investigated, when\ngenerating bootstrap samples\nfrom $\\mathbb{F}_n$ and $\\tilde{F}_n$. It is shown that both the\nmethods lead to consistent estimators under mild conditions on $m$. In\nSection \\ref{discussion}, we discuss our findings, especially the\nnonconvergence and its implications. The\n\\hyperref[app]{Appendix}, provides the details of some arguments used\nin proving the\nmain results.\n\n\\section{Uniform convergence}\\label{prelim}\n\nFor the rest of the paper, $F$ denotes a distribution function with\n$F(0) = 0$ and a density $f$ that is nonincreasing on $[0,\\infty)$ and\ncontinuously differentiable near $t_0 \\in(0,\\infty)$ with nonzero\nderivative $f'(t_0) < 0$. If $g\\dvtx I \\to{\\mathbb R}$ is a bounded\nfunction, write $\\Vert g\\Vert:= {\\sup_{x\\in I}} |g(x)|$. Next, let $F_n$\nbe distribution functions with $F_n(0) = 0$, that converge weakly to\n$F$ and, therefore,\n\\begin{equation}\n\\label{eq:cndtn0}\n{\\lim_{n\\to\\infty}} \\| F_n-F \\| = 0.\n\\end{equation}\nLet $X_{n,1},X_{n,2},\\ldots,X_{n,m_n} \\stackrel{\\mathrm{ind}}{\\sim}\nF_n$, where $m_n\n\\le n$ is a nondecreasing sequence of integers for which $m_n \\to\\infty\n$; let ${\\mathbb F}_{n,m_n}$ denote the EDF of $X_{n,1},X_{n,2},\n\\ldots,\\break X_{n,m_n}$; and let\n\\[\n\\Delta_n := m_n^{1\/3} \\{\\tilde f_{n,m_n}(t_0) - f_n(t_0)\\},\n\\]\nwhere $\\tilde f_{n,m_n}(t_0)$ is the Grenander estimator computed from\n$X_{n,1},X_{n,2},\\ldots,\\break X_{n,m_n}$ and $f_n(t_0)$ is the density of\n$F_n$ at $t_0$ or a surrogate. Next, let $I_m = [-t_0m^{1\/3},\\infty\n)$ and\n\\begin{equation}\\label{eq:Z_process}\n\\quad \\mathbb{Z}_n(h) := m_n^{2\/3} \\{ \\mathbb{F}_{n,m_n}(t_0 +\nm_n^{-{1\/3}}h) - \\mathbb{F}_{n,m_n}(t_0) - f_n(t_0) m_n^{-{1\/3}}h \\}\n\\end{equation}\nfor $h \\in I_{m_n}$ and observe that $\\Delta_n$ is the left-hand\nderivative at $0$ of the least concave majorant of ${\\mathbb Z}_n$. It\nis fairly easy to obtain the asymptotic distribution of $\\mathbb{Z}_n$.\nThe asymptotic distribution of $\\Delta_n$ may then be obtained from the\nContinuous Mapping theorem. Stochastic processes are regarded as random\nelements in $D(\\mathbb{R})$, the space of right continuous functions on\n$\\mathbb{R}$ with left limits, equipped with the projection $\\sigma\n$-field and the topology of uniform convergence on compacta. See\nPollard (\\citeyear{polllard84}), Chapters IV and V for background.\n\n\\subsection{Convergence of ${\\mathbb Z}_n$}\n\nIt is convenient to\ndecompose $\\mathbb{Z}_n$ into the sum of $\\mathbb{Z}_{n,1}$ and $\\mathbb\n{Z}_{n,2}$ where\n\\begin{eqnarray*}\n\\mathbb{Z}_{n,1}(h) &:=& m_n^{2\/3} \\{ (\\mathbb\n{F}_{n,m_n} - F_{n})(t_0 + m_n^{-{1\/3}}h) - (\\mathbb{F}_{n,m_n}-\nF_{n})(t_0) \\},\\\\\n\\mathbb{Z}_{n,2}(h) &:=& m_n^{2\/3} \\{\nF_{n}(t_0 + m_n^{-{1\/3}} h)- F_{n}(t_0) - f_n(t_0) m_n^{-{1\/3}}h \\}.\n\\end{eqnarray*}\nObserve that ${\\mathbb Z}_{n,2}$ depends only on $F_n$ and $f_n$; only\n${\\mathbb Z}_{n,1}$ depends on $X_{n,1},\\ldots,\\break X_{n,m_n}$. Let\n$\\mathbb{W}_1$ be a standard two-sided Brownian motion on $\\mathbb{R}$\nwith $\\mathbb{W}_1(0) = 0$, and $\\mathbb{Z}_{1}(h) = \\mathbb{W}_1 [f(t_0)h]$.\n\\begin{prop}\\label{prop:Z_conv}\nIf\n\\begin{equation}\n\\label{eq:cndtn1}\n\\lim_{n\\to\\infty} m_n^{1\/3} | F_n(t_0+m_n^{-{1\/3}}h) -\nF_n(t_0) - f(t_0) m_n^{-{1\/3}}h | = 0\n\\end{equation}\nuniformly on compacts (in $h$), then ${\\mathbb Z}_{n,1} \\Rightarrow\n{\\mathbb Z}_1$; and if there is a continuous\nfunction ${\\mathbb Z}_2$ for which\n\\begin{equation}\n\\label{eq:cndtn2}\n\\lim_{n\\to\\infty} {\\mathbb Z}_{n,2}(h) = {\\mathbb Z}_2(h)\n\\end{equation}\nuniformly on compact intervals, then ${\\mathbb Z}_n \\Rightarrow\n{\\mathbb Z} := {\\mathbb Z}_1 + {\\mathbb Z}_2$.\n\\end{prop}\n\\begin{pf}\nThe Hungarian Embedding theorem of K\\'omlos, Major and\\break  Tusn\\'ady\n(\\citeyear{kmt75})\nis used. We may suppose that\n$X_{n,i} = F_n^{\\#}(U_{i})$, where $F_n^{\\#}(u) = \\inf\\{x\\dvtx F_n(x)\n\\ge\nu\\}$ and $U_{1},U_2,\\ldots$ are i.i.d. Uniform$(0,1)$ random variables.\nLet $\\mathbb{U}_n$ denote the EDF of $U_{1},\\ldots, U_{n}$, $\\mathbb\n{E}_n(t) = \\sqrt{n} [\\mathbb{U}_n(t) - t]$, and $\\mathbb{V}_n = \\sqrt\n{m_n} (\\mathbb{F}_{n,m_n} - F_n)$. Then $\\mathbb{V}_n = \\mathbb\n{E}_{m_n} \\circ F_n$. By Hungarian Embedding, we may also suppose that\nthe probability space supports a sequence of Brownian Bridges $\\{\\mathbb\n{B}_n^0\\}_{n \\ge1}$ for which\n\\begin{equation}\n\\label{eq:kmt}\n{\\sup_{0 \\le t \\le1}} | \\mathbb{E}_{n}(t) - \\mathbb{B}_n^0(t)| = O\n\\biggl[{\\log(n)\\over\\sqrt{n}} \\biggr] \\qquad\\mbox{a.s.},\n\\end{equation}\nand a standard normal random variable $\\eta$ that is independent of $\\{\n\\mathbb{B}_n^0\\}_{n \\ge1}$. Define a version $\\mathbb{B}_n$ of\nBrownian motion by $\\mathbb{B}_n(t) = \\mathbb{B}_n^0(t) + \\eta t$, for\n$t \\in[0,1]$. Then\n\\begin{eqnarray}\\label{eq:boots_proc1}\n\\mathbb{Z}_{n,1} (h) & = & m_n^{1\/6} \\{\\mathbb\n{E}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{E}_{m_n}[F_{n}(t_0)] \\} \\nonumber\\\\[-8pt]\\\\[-8pt]\n& = & m_n^{1\/6} \\{\\mathbb{B}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{B}_{m_n}[F_n(t_0)] \\} + \\mathbb{R}_{n}(h),\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n|\\mathbb{R}_n(h)| & \\le& 2 m_n^{1\/6} {\\sup_{0\n\\le t \\le1}} |\\mathbb{E}_{m_n}(t) - \\mathbb{B}_{m_n}^0(t)| \\\\\n&&{} + m_n^{1\/6}|\\eta| |F_n(t_0 + m_n^{-{1\/3}}h) - F_n(t_0)|\n\\rightarrow0\n\\end{eqnarray*}\nuniformly on compacta w.p. 1 using (\\ref{eq:cndtn1}) and (\\ref\n{eq:kmt}). Let\n\\[\n\\mathbb{X}_n(h) := m_n^{1\/6}\\{\\mathbb\n{B}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{B}_{m_n}[F_n(t_0)] \\}\n\\]\nand observe that $\\mathbb{X}_n$ is a mean zero Gaussian process defined\non $I_{m_n}$ with independent increments and\ncovariance kernel\n\\[\nK_n(h_1,h_2) = m_n^{1\/3} | F_n[t_0 + \\operatorname{sign} \\{h_1\\}\nm_n^{-{1\/3}}(|h_1| \\wedge|h_2|)] - F_n(t_0) | \\mathbf{1}\\{\nh_1 h_2 > 0\\}.\n\\]\nIt now follows from Theorem V.19 in Pollard (\\citeyear{polllard84}) and\n(\\ref\n{eq:cndtn1}) that $\\mathbb{X}_n(h)$ converges in\ndistribution to $\\mathbb{W}_1[f(t_0)h]$ in $D([-c,c])$ for every $c >\n0$, establishing the first assertion of the\nproposition. The second then follows from Slutsky's theorem.\n\\end{pf}\n\n\\subsection{Convergence of $\\Delta_n$}\n\nUnfortunately, $\\Delta_n$ is not\nquite a continuous functional of ${\\mathbb\nZ}_n$. If $f\\dvtx I \\to{\\mathbb R}$, write $f|J$ to denote the restriction\nof $f$ to $J \\subseteq I$; and if $I$ and $J$ are intervals and $f$ is\nbounded, write $L_Jf$ for the least concave majorant of the\nrestriction. Thus, $\\tilde{F}_n = L_{[0,\\infty)}{\\mathbb F}_n$ in the\n\\hyperref[intro]{Introduction}.\n\\begin{lemma}\\label{lem:ww} Let $I$ be a closed interval; let $f\\dvtx I\n\\to\n{\\mathbb R}$ be a bounded upper semi-continuous function on $I$; and\nlet $a_1,a_2,b_1,b_2 \\in I$ with $b_1 < a_1 < a_2 < b_2$. If\n$2f[{1\\over2}(a_i+b_i)] > L_{I}\nf(a_i) + L_{I} f(b_i), i = 1,2$, then $L_If(x) = L_{[b_1,b_2]}f(x)$ for\n$a_1 \\le x \\le a_2$.\n\\end{lemma}\n\\begin{pf} This follows from the proof of Lemmas 5.1 and 5.2 of\nWang and Woodroofe (\\citeyear{wangWoo07}). In that lemma\ncontinuity was assumed, but only upper semi-continuity was used in the\n(short) proof.\n\\end{pf}\n\nRecall Marshall's lemma: if $I$ is an interval, $f: I \\to{\\mathbb R}$\nis bounded, and $g\\dvtx I \\to{\\mathbb R}$ is concave, then $\\Vert\nL_If-g\\Vert\\le\\Vert f-g\\Vert$. See, for example, Robertson, Wright\nand Dykstra [(\\citeyear{RWD88}), page 329] for a proof. Write $\\tilde\n{F}_{n,m_n} =\nL_{[0,\\infty)}{\\mathbb F}_{n,m_n}$.\n\\begin{lemma}\\label{lem:loc}\nIf $\\delta> 0$ is so small that $F$ is strictly concave on\n$[t_0-2\\delta,t_0+2\\delta]$ and (\\ref{eq:cndtn0}) holds then ${\\tilde\nF}_{n,m_n} = L_{[t_0-2\\delta,t_0+2\\delta]}{\\mathbb F}_{n,m_n}$ on\n$[t_0-\\delta,t_0+\\delta]$ for all large $n$ w.p. 1.\n\\end{lemma}\n\\begin{pf} Since $F$ is strictly concave on $[t_0-2\\delta,t_0+2\\delta\n], 2F(t_0\\pm{3\\over2}\\delta) >\nF(t_0\\pm\\delta) + F(t_0\\pm2\\delta)$. Then\n\\begin{eqnarray*}\n\\| \\tilde{F}_{n,m_n} - F \\| & \\le& \\|{\\mathbb F}_{n,m_n} - F \\|\n\\\\\n& \\le& \\|{\\mathbb F}_{n,m_n} - F_n \\| + \\|F_n - F\\| \\\\\n& \\le& \\frac{1}{\\sqrt{m_n}} \\|\\mathbb{E}_{m_n}\\| + \\|F_n - F \\| \\to0\n\\qquad\\mbox{w.p. } 1\n\\end{eqnarray*}\nby Marshall's lemma, (\\ref{eq:cndtn0}) and the Glivenko--Cantelli\ntheorem. Thus,\\break $2{\\mathbb F}_{n,m_n}(t_0\\pm\n{3\\over2}\\delta) > \\tilde{F}_{n,m_n}(t_0\\pm\\delta) + \\tilde\n{F}_{n,m_n}(t_0\\pm2\\delta)$, for all large $n$ w.p. 1, and Lemma \\ref\n{lem:loc} follows from Lemma \\ref{lem:ww}.\n\\end{pf}\n\\begin{prop}\\label{prop:loc} \\textup{(i)} Suppose that (\\ref{eq:cndtn0}) and\n(\\ref{eq:cndtn1}) hold and given $\\gamma> 0$, there are $0 < \\delta<\n1$ and $C > 0$ for which\n\\begin{equation}\n\\label{eq:cndtn3}\n\\bigl| F_n(t_0+h) - F_n(t_0) - f_n(t_0)h - \\tfrac{1}{2} f'(t_0) h^2\n\\bigr| \\le\\gamma h^2 + C m_n^{-{2\/3}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:cndtn4}\n| F_n(t_0+h) - F_n(t_0) | \\le C (|h| + m_n^{-{1\/3}})\n\\end{equation}\nfor $|h| \\le\\delta$ and for all large $n$. If $J$ is a compact\ninterval and $\\varepsilon>0$, then there is a compact $K \\supseteq J$,\ndepending only on $\\varepsilon, J, C, \\gamma$, and $\\delta$, for which\n\\begin{equation}\n\\label{eq:loc}\nP [L_{I_{m_n}}{\\mathbb Z}_n = L_K{\\mathbb Z}_n \\mbox{ on } J ]\n\\ge1 - \\varepsilon\n\\end{equation}\nfor all large $n$.\n\n\\textup{(ii)} Let $\\mathbb{Y}$ be an a.s. continuous stochastic process on\n$\\mathbb{R}$ that is a.s. bounded above. If $\\lim_{|h| \\rightarrow\n\\infty} \\mathbb{Y}(h)\/|h| = -\\infty$ a.e., then the compact $K\n\\supseteq J$ can be chosen so that\n\\begin{equation}\n\\label{eq:loc2}\nP [L_{\\mathbb R}{\\mathbb Y} = L_K{\\mathbb Y} \\mbox{ on } J ]\n\\ge1 - \\varepsilon.\n\\end{equation}\n\\end{prop}\n\\begin{pf} For a fixed sequence ($F_n \\equiv F$) (\\ref{eq:loc}) would\nfollow from the assertion in Example 6.5 of Kim and Pollard (\\citeyear\n{kimPo90}), and\nit is possible to adapt their argument to a triangular array using (\\ref\n{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) in place of Taylor series expansion. A different\nproof is presented in the \\hyperref[app]{Appendix}.\n\\end{pf}\n\nWe will use the following easily verified fact. In its statement, the\nmetric space $\\mathcal{X}$ is to be endowed with\nthe projection $\\sigma$-field. See Pollard (\\citeyear{polllard84}),\npage 70.\n\\begin{lemma}\\label{lemma:con_derv} Let $\\{X_{n,c}\\}, \\{Y_n\\}, \\{W_c\\}$\nand $Y$ be sets of random elements taking values in a metric space\n$(\\mathcal{X}$,$d)$, $n=0,1,\\ldots,$ and $c \\in\\mathbb{R}$. If for any\n$\\delta> 0$,\n\\begin{longlist}\n\\item $\\lim_{c \\rightarrow\\infty} \\limsup_{n \\rightarrow\\infty}\nP\\{d(X_{n,c},Y_n) > \\delta\\} = 0$,\n\n\\item $\\lim_{c \\rightarrow\\infty} P\\{d(W_{c},Y) > \\delta\\} = 0$,\n\n\\item $X_{n,c} \\Rightarrow W_c$ as $n \\rightarrow\\infty$ for\nevery $c \\in\\mathbb{R}$,\n\\end{longlist}\nthen $Y_n \\Rightarrow Y$ as $n \\rightarrow\\infty$.\n\\end{lemma}\n\\begin{cor}\\label{cor:convdelta} If (\\ref{eq:loc}) and (\\ref{eq:loc2})\nhold, and $\\mathbb{Z}_n \\Rightarrow\\mathbb{Y}$, then\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Y}$ in\n$D({\\mathbb R})$ and $ \\Delta_n \\Rightarrow(L_{\\mathbb R}{\\mathbb Y})'(0)$.\n\\end{cor}\n\\begin{pf} It suffices to show that $L_{I_{m_n}}{\\mathbb Z}_n|J\n\\Rightarrow L_{\\mathbb R}{\\mathbb Y}|J$ in $D(J)$, for every compact\ninterval $J \\subseteq\\mathbb{R}$. Given $J$ and $\\varepsilon> 0$, there\nexists $K_\\varepsilon$, a compact, $K_\\varepsilon\\supseteq J$, such that\n(\\ref{eq:loc}) and (\\ref{eq:loc2}) hold. This verifies (i) and (ii)\nof Lemma \\ref{lemma:con_derv} with $c = 1\/\\varepsilon$, $X_{n,c}=\nL_{K_\\varepsilon}{\\mathbb Z}_n$, $Y_n = L_{I_{m_n}}{\\mathbb Z}_n$, $W_c =\nL_{K_\\varepsilon}{\\mathbb Y}$, $Y = L_{\\mathbb R}{\\mathbb Y}$ and $d(x,y)\n= \\sup_{t \\in J} |x(t) - y(t)|$. Clearly, $L_{K_\\varepsilon}{\\mathbb\nZ}_n|J \\Rightarrow L_{K_\\varepsilon}{\\mathbb Y}|J$ in $D(J)$, by the\nContinuous Mapping theorem, verifying condition (iii). Thus,\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Y}$ in\n$D(\\mathbb R)$. Another application of the Continuous Mapping theorem\n[via the lemma on page 330 of Robertson, Wright and Dykstra (\\citeyear\n{RWD88})] in\nconjunction with (\\ref{eq:loc}), (\\ref{eq:loc2}) and Lemma \\ref\n{lemma:con_derv} then shows that $\\Delta_n = (L_{I_{m_n}}{\\mathbb\nZ}_n)'(0) \\Rightarrow(L_{\\mathbb R}{\\mathbb Y})'(0)$.\n\\end{pf}\n\\begin{cor}\\label{cor:convdelta2} If (\\ref{eq:cndtn0}), (\\ref\n{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) hold and\n\\[\n\\lim_{|h| \\rightarrow\\infty}\n\\mathbb{Z}(h)\/|h| = -\\infty,\n\\]\nthen\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Z}$ in\n$D({\\mathbb R})$ and $ \\Delta_n \\Rightarrow\n(L_{\\mathbb R}{\\mathbb Z})'(0)$; and if $\\mathbb{Z}_{2}(h) = f'(t_0)\nh^2\/2$, then $ \\Delta_n \\Rightarrow2 |\\frac{1}{2} f(t_0) f'(t_0)|^{1\/3}\n\\mathbb C$, where $\\mathbb C$ has Chernoff's distribution.\n\\end{cor}\n\\begin{pf} The convergence follows directly from Proposition \\ref\n{prop:loc} and\\break Corollary~\\ref{cor:convdelta}. Note that if $\\mathbb\n{Z}_{2}(h) = f'(t_0) h^2\/2$, then (\\ref{eq:loc}) and (\\ref{eq:loc2})\nhold\\break and~Corollary~\\ref{cor:convdelta} can be applied. That\n$(L_{\\mathbb R}{\\mathbb Z})'(0)$ is distributed as\\break $2 |\\frac{1}{2}\nf(t_0) f'(t_0)|^{1\/3} \\mathbb C$ when $\\mathbb{Z}_{2}(h) =\nf'(t_0) h^2\/2$ follows from elementary properties of Brownian motion\nvia the ``switching'' argument of Groeneboom (\\citeyear{groene85}).\n\\end{pf}\n\n\\subsection{Remarks on the conditions}\\label{remarks}\n\nIf $F_n \\equiv F$\nand $f_n \\equiv f$, then clearly (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}),\n(\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and (\\ref{eq:cndtn4}) all hold\nwith ${\\mathbb Z}_2(h) = f'(t_0)h^2\/2$ for some $0 < \\delta< 1$ and $C\n\\ge f(t_0 - \\delta)$ by a Taylor expansion of $F$ and the continuity of\n$f$ and $f'$ around $t_0$.\n\\begin{cor}\\label{cor:smoothF} If there is a $\\delta> 0$ for which\n$F_n$ has a continuously differentiable density\n$f_n$ on $[t_0 - \\delta, t_0 + \\delta]$, and\n\\begin{equation}\n\\label{eq:cndsmoothF}\\quad\n\\lim_{n \\rightarrow\\infty} \\Bigl[ \\|\nF_n - F\\| + \\sup_{|t - t_0| < \\delta} \\bigl(\n|f_n(t) - f(t)| + |f_n'(t) - f'(t)| \\bigr) \\Bigr] = 0,\n\\end{equation}\nthen (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref\n{eq:cndtn3}) and (\\ref{eq:cndtn4}) hold with\n${\\mathbb Z}_2(h) = f'(t_0)h^2\/2$, and $\\Delta_n \\Rightarrow2 |\\frac\n{1}{2} f(t_0) f'(t_0)|^{1\/3} \\mathbb C$.\n\\end{cor}\n\\begin{pf} The result can be immediately derived from Taylor expansion\nof $F_n$ and the continuity of $f$ and $f'$\naround $t_0$. To illustrate the idea, we show that (\\ref{eq:cndtn3})\nholds. Let $\\gamma> 0$ be given. Clearly,\n\\begin{eqnarray}\\label{sm_boot_t6}\n&&\\biggl| F_n(t_0 + h) - F_n(t_0) - f_n(t_0) h - \\frac{1}{2} h^2 f'(t_0)\n\\biggr| \\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\le{\\frac{1}{2} h^2 \\sup_{|s| \\le|h|}} |f_n'(t_0 + s) -\nf'(t_0)|.\\nonumber\n\\end{eqnarray}\nLet $\\delta> 0$ be so small that $|f'(t) - f'(t_0)| \\le\\gamma$ for\n$|t - t_0| < \\delta$, and let $n_0$ be so large that ${\\sup_{|t - t_0|\n\\le\\delta} }|f_n'(t) - f'(t)| \\le\\gamma$ for $n \\ge n_0$. Then the\nlast line in\n(\\ref{sm_boot_t6}) is at most $\\gamma h^2$ for $|h| \\le\\delta$ and $n\n\\ge n_0$.\n\\end{pf}\n\nAnother useful remark, used below, is that if $\\lim_{n \\rightarrow\n\\infty} m_n^{1\/3} \\|F_{m_n} - F\\| = 0$, then (\\ref{eq:cndtn0}),\n(\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4}) hold.\n\nIn the next three sections, we apply Proposition \\ref{prop:Z_conv} and\nCorollary \\ref{cor:convdelta} to bootstrap samples drawn from the EDF,\nits LCM, and smoothed versions thereof. Thus, let $X_1,X_2,\\ldots\n\\stackrel{\\mathrm{ind}}{\\sim} F$; let ${\\mathbb F}_n$ be the EDF of\n$X_1,\\ldots,X_n$; and\nlet $\\tilde{F}_n$ be its LCM. If $F_n = {\\mathbb F}_n$, then\n(\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4}) hold almost\nsurely by the above remark, since\n\\begin{equation}\n\\label{eq:lil}\n\\Vert{\\mathbb F}_n - F\\Vert= O \\Biggl[\\sqrt{\\log\\log(n)\\over n}\n\\Biggr]\\qquad\\mbox{a.s.}\n\\end{equation}\nby the Law of the Iterated Logarithm for the EDF, which may be deduced\nfrom Hungarian Embedding; and the same is true if $F_n = \\tilde{F}_n$\nsince $\\Vert\\tilde{F}_n-F\\Vert\\le\\Vert{\\mathbb F}_n-F\\Vert$, by\nMarshall's lemma.\n\nIf $m_n = n$ and $f_n = \\tilde{f}_n$, then (\\ref{eq:cndtn2}) is not\nsatisfied almost surely or in probability by either ${\\mathbb F}_n$ or\n$\\tilde{F}_n$. For either choice, (\\ref{eq:cndtn3}) is satisfied in\nprobability if $f_n = f$.\n\\begin{prop}\\label{prop:edflcm}\nSuppose that $m_n = n$ and that $f_n = f$. If $F_n$ is either the EDF\n${\\mathbb F}_n$ or its LCM $\\tilde{F}_n$, then\nfor any $\\gamma, \\varepsilon> 0$, there are $C > 0$ and $0 < \\delta< 1$\nfor which (\\ref{eq:cndtn3}) holds with\nprobability at least $1- \\varepsilon$ for all large $n$.\n\\end{prop}\n\nThe proof is included in the \\hyperref[app]{Appendix}.\n\n\\section{Inconsistency and nonconvergence of the bootstrap}\\label{boots_prob}\n\nWe begin with a brief discussion of the bootstrap.\n\n\\subsection{Generalities}\\label{generalities}\n\nNow, suppose that $X_1,X_2,\\ldots\\stackrel{\\mathrm{ind}}{\\sim} F$ are\ndefined on\na probability space $(\\Omega,\\mathcal{A}, P)$. Write ${\\mathbf X}_n =\n(X_1,\\ldots,X_n)$ and suppose that the distribution function, $H_n$\nsay, of the random variable $R_n(\\mathbf{X}_n,F)$ is of interest. The\nbootstrap methodology can be broken into three simple steps:\n\n\\begin{longlist}\n\\item Construct an estimator $\\hat{F}_n$ of $F$ from ${\\mathbf X}_n$;\n\n\\item let $X_1^{*},\\ldots,X_{m_n}^{*} \\stackrel{\\mathrm{ind}}{\\sim\n}\\hat{F}_n$ be\nconditionally i.i.d. given ${\\mathbf X}_n$;\n\n\\item then let ${\\mathbf X}_n^{*} = (X_1^{*},\\ldots,X_{m_n}^{*})$\nand estimate $H_n$ by the conditional\ndistribution function of $R_n^{*} = R({\\mathbf X}_n^{*},\\hat{F}_n)$\ngiven ${\\mathbf X}_n$; that is\n\\[\nH_{n}^*(x) = P^*\\{R^*_n \\le x\\},\n\\]\nwhere $P^*\\{\\cdot\\}$ is the conditional probability given the data\n$\\mathbf{X}_n$, or equivalently, the entire sequence $\\mathbf{X} =\n(X_1,X_2,\\ldots)$.\n\\end{longlist}\nChoices of $\\hat{F}_n$ considered below are the EDF ${\\mathbb F}_n$,\nits least concave majorant\n$\\tilde{F}_n$, and smoothed versions thereof.\n\nLet $d$ denote the Levy metric or any other metric metrizing weak\nconvergence of distribution functions. We say that\n$H_{n}^*$ is \\textit{weakly}, \\textit{respectively}, \\textit{strongly},\n\\textit{consistent} if\n$d(H_n,H_n^*)\\stackrel{P}{\\rightarrow} 0$, respectively, $d(H_n,H_n^{*})\n\\to0$ a.s. If $H_{n}$ has a weak limit $H$, then consistency requires\n$H_{n}^*$ to converge weakly to $H$, in probability; and if $H$ is\ncontinuous, consistency requires\n\\[\n{\\sup_{x \\in\\mathbb{R}}} |H_{n}^*(x) - H(x)| \\stackrel{P}{\\rightarrow} 0\n\\qquad\\mbox{as } n \\rightarrow\\infty.\n\\]\nThere is also the apparent possibility that $H_n^{*}$ could converge to\na random limit; that is, that there is a\n$G\\dvtx\\Omega\\times\\mathbb{R} \\to[0,1]$ for which $G(\\omega,\\cdot)$ is a\ndistribution function for each $\\omega\\in\n\\Omega$, $G(\\cdot,x)$ is measurable for each $x \\in\\mathbb{R}$, and\n$d(G,H_n^{*}) \\stackrel{P}{\\rightarrow} 0$. This\npossibility is only apparent, however, if $\\hat{F}_n$ depends only on\nthe order statistics. For if $h$ is a bounded\ncontinuous function on $\\mathbb{R}$, then any limit in probability of\n$\\int_{\\mathbb{R}} h(x) H_n^*(\\omega;dx)$ must be invariant under\nfinite permutations of $X_1,X_2,\\ldots$ up to equivalence, and thus,\nmust be almost surely constant by the Hewitt--Savage zero--one law\n[Breiman (\\citeyear{breiman68})]. Let $\\bar G(x) = \\int_{\\Omega}\nG(\\omega;x) P(d\\omega\n)$. Then $\\bar G$ is a distribution function and $\\int_{\\mathbb{R}}\nh(x) G(\\omega;dx) = \\int_{\\mathbb{R}} h(x) \\bar G(dx)$ a.s. for each\nbounded continuous~$h$, and therefore for any countable collection of\nbounded continuous $h$. It follows that $G(\\omega;x) =\\bar G(x)$ a.e.\n$\\omega$ for all $x$ by letting $h$ approach indicator functions.\n\nNow let\n\\[\n\\Delta_n = n^{1\/3} \\{\\tilde f_n(t_0) - f(t_0) \\}\n\\quad\\mbox{and}\\quad \\Delta_n^* = m_n^{1\/3}\n\\{\\tilde f_{n,m_n}^*(t_0) - \\hat f_n(t_0) \\},\n\\]\nwhere $\\hat f_n(t_0)$ is an estimate of $f(t_0)$, for example, $\\tilde\nf_n(t_0)$, and $\\tilde f_{n,m_n}^*(t_0)$ is the\nGrenander estimator computed from the bootstrap sample $X_1^{*},\\ldots\n,X_{m_n}^{*}$. Then weak (strong) consistency of the bootstrap means\n\\begin{equation}\n\\label{eq:boot_limit}\n{\\sup_{x \\in\\mathbb R}} |P^*[\\Delta_n^* \\le x] - P[\\Delta_n \\le x]|\n\\rightarrow0\n\\end{equation}\nin probability (almost surely), since the limiting distribution (\\ref\n{eq:chrnff}) of $\\Delta_n$ is continuous.\n\n\\subsection{Bootstrapping from the NPMLE $\\tilde{F}_n$}\\label{bootsNPMLE}\n\nConsider now the case in which $m_n = n$, $\\hat F_n = \\tilde{F}_n$, and\n$\\hat{f}_n(t_0) = \\tilde{f}_n(t_0)$. Let\n\\[\n\\mathbb{Z}_n^*(h) := n^{2\/3} \\{ \\mathbb{F}_n^*(t_0 +\nn^{-{1\/3}}h) -\n\\mathbb{F}_n^*(t_0) - \\tilde f_n(t_0)n^{-{1\/3}}h \\}\n\\]\nfor $h \\in I_n = [-n^{1\/3}t_0, \\infty)$, where $\\mathbb{F}_{n}^*$\nis the EDF of the bootstrap sample\n$X_1^*,\\ldots,X_n^* \\sim\\tilde F_n$. Then $\\mathbb{Z}_n^* = \\mathbb\n{Z}_{n,1}^* + \\mathbb{Z}_{n,2}$, where\n\\begin{eqnarray}\n\\mathbb{Z}_{n,1}^* (h) &=& n^{2\/3} \\{ (\\mathbb{F}_n^*-\\tilde\n{F}_n)(t_0 + n^{-{1\/3}}h) -\n(\\mathbb{F}_n^*-\\tilde{F}_n)(t_0) \\},\n\\\\\n\\mathbb{Z}_{n,2}(h) &=& n^{2\/3} \\{\\tilde{F}_n(t_0 + h\nn^{-{1\/3}}) - \\tilde{F}_n(t_0) - \\tilde f_n(t_0) n^{-{1\/3}}h \\}.\n\\end{eqnarray}\nFurther, let $\\mathbb{W}_1$ and $\\mathbb{W}_2$ be two independent\ntwo-sided standard Brownian motions on $\\mathbb{R}$\nwith $\\mathbb{W}_1(0) = \\mathbb{W}_2(0) = 0$,\n\\begin{eqnarray*}\n\\mathbb{Z}_{1}(h) &=& \\mathbb{W}_1[f(t_0)h],\\\\\n\\mathbb{Z}_{2}^0(h) &=& \\mathbb{W}_2[f(t_0)h] +\n\\tfrac{1}{2}f'(t_0)h^2,\\\\\n\\mathbb{Z}_{2}(h) &=& L_{\\mathbb R} \\mathbb{Z}_{2}^0(h) - L_{\\mathbb R}\n\\mathbb{Z}_{2}^0(0) - (L_{\\mathbb{R}}\n\\mathbb{Z}_2^0)'(0)h,\\\\\n\\mathbb{Z} &=& \\mathbb{Z}_1 + \\mathbb{Z}_{2}.\n\\end{eqnarray*}\nThen $\\Delta_n^*$ equals the left derivative at $h = 0$ of the LCM of\n$\\mathbb{Z}_n^*$. It is first shown that\n$\\mathbb{Z}_n^*$ converges in distribution to $\\mathbb{Z}$ but the\nconditional distributions of $\\mathbb{Z}_n^*$ do not have a limit. The\nfollowing two lemmas are needed.\n\\begin{lemma}\\label{lemma:independence} Let $W_n$ and $W_n^*$ be random\nvectors in $\\mathbb{R}^l$ and $\\mathbb{R}^k$,\nrespectively; let $Q$ and $Q^*$ denote distributions on the Borel sets\nof $\\mathbb{R}^l$ and $\\mathbb{R}^k$; and let\n$\\mathcal{F}_n$ be sigma-fields for which $W_n$ is $\\mathcal\n{F}_n$-measurable. If the distribution of $W_n$ converges to $Q$ and\nthe conditional distribution of $W_n^*$ given $\\mathcal{F}_n$ converges\nin probability to $Q^*$, then the joint distribution of $(W_n,W_n^*)$\nconverges to the product measure $Q \\times Q^*$.\n\\end{lemma}\n\\begin{pf} The above lemma can be proved easily using characteristic\nfunctions. Kosorok (\\citeyear{kosorok07}) includes a detailed proof.\n\\end{pf}\n\nThe next lemma uses a special case of the Convergence of Types theorem\n[Lo\\`{e}ve (\\citeyear{loeve63}), page 203]: let $V, W, V_n$ be random\nvariables and\n$b_n$ be constants; if $V$ has a nondegenerate distribution, $V_n\n\\Rightarrow V$ as $n \\to\n\\infty$, and $V_n + b_n \\Rightarrow W$, then $b = \\lim_{n \\to\\infty}\nb_n$ exists and $W$ has the same distribution as $V+b$.\n\\begin{lemma}\\label{lemma:emp_boot1} Let $\\mathbf{X}_n^*$ be a\nbootstrap sample generated from the data $\\mathbf{X}_n$. Let $Y_n :=\n\\psi_n(\\mathbf{X}_n)$ and $Z_n := \\phi_n(\\mathbf{X}_n,\\mathbf{X}_n^*)$\nwhere $\\psi_n\\dvtx\\mathbb{R}^n \\to\\mathbb{R}$ and $\\phi_n\\dvtx\\mathbb{R}^{2\nn} \\to\\mathbb{R}$ are measurable functions; and let $K_n$ and $L_n$ be\nthe conditional distribution functions of $Y_n + Z_n$ and $Z_n$ given\n${\\mathbf X}_n$, respectively. If there are distribution functions $K$\nand $L$ for which $L$ is nondegenerate, $d(K_n,K) \\stackrel\n{P}{\\rightarrow} 0$ and $d(L_n,L) \\stackrel{P}{\\rightarrow} 0$ then\nthere is a random variable $Y$ for which $Y_n \\stackrel{P}{\\rightarrow} Y$.\n\\end{lemma}\n\\begin{pf} If $\\{n_k\\}$ is any subsequence, then there exists a further\nsubsequence $\\{n_{k_l}\\}$ for which $d(K_{n_{k_l}},K) \\rightarrow 0$\na.s. and $d(L_{n_{k_l}},L) \\rightarrow 0$ a.s. Then $Y := \\lim_{l\n\\rightarrow\\infty} Y_{n_{k_l}}$ exists a.s. by the Convergence of\nTypes theorem,\napplied conditionally given ${\\mathbf X} := (X_1,X_2,\\ldots)$ with\n$b_l = Y_{n_{k_l}}$. Note that $Y$ does not depend on the subsequence\n$n_{k_l}$, since two such subsequences can be joined to form another\nsubsequence using which we can argue the uniqueness.\n\\end{pf}\n\\begin{theorem}\\label{thm:bootmle}\n\\textup{(i)} The conditional distribution of $\\mathbb{Z}_{n,1}^*$ given\n$\\mathbf{X} = (X_1,\\break X_2,\\ldots)$ converges a.s. to the distribution\nof $\\mathbb{Z}_1$.\n\n{\\smallskipamount=0pt\n\\begin{longlist}[(iii)]\n\\item[(ii)] The unconditional distribution of $\\mathbb{Z}_{n,2}$\nconverges to that of $\\mathbb{Z}_2$ and the\nunconditional distributions of $(\\mathbb{Z}_{n,1}^*,\\mathbb{Z}_{n,2})$,\nand $\\mathbb{Z}_n^*$ converge to those of\n$(\\mathbb{Z}_{1}, \\mathbb{Z}_{2})$ and~$\\mathbb{Z}$.\n\n\\item[(iii)] The unconditional distribution of $\\Delta_n^*$ converges\nto that of $(L_{\\mathbb{R}}\\mathbb{Z})'(0)$, and (\\ref{eq:boot_limit}) fails.\n\n\\item[(iv)] Conditional on $\\mathbf{X}$, the distribution of $\\mathbb\n{Z}_n^*$ does not have a weak limit in\nprobability.\n\n\\item[(v)] If the conditional distribution function of $\\Delta_n^*$\nconverges in probability, then $(L_{\\mathbb{R}} {\\mathbb{Z}})'(0)$ and\n${\\mathbb Z}_2$ must be independent.\n\\end{longlist}}\n\\end{theorem}\n\\begin{pf} (i) The conditional convergence of $\\mathbb\n{Z}_{n,1}^*$ follows from Proposition \\ref{prop:Z_conv} with $m_n = n$,\n$F_n = \\tilde{F}_n$, $\\mathbb{F}_{n,m_n} = \\mathbb{F}_n^*$, applied\nconditionally given ${\\mathbf X}$. It is only necessary to show that\n(\\ref{eq:cndtn1}) holds a.s., and this follows from the Law of the\nIterated Logarithm for ${\\mathbb F}_n$ and Marshall's lemma, as\nexplained in Section \\ref{remarks}. The unconditional limiting\ndistribution of\n$\\mathbb{Z}_{n,1}^*$ must also be that of $\\mathbb{Z}_1$.\n\n(ii) Let\n\\[\n{\\mathbb Z}_{n,2}^0(h) = n^{2\/3}[{\\mathbb F}_n(t_0+n^{-{1\/3}}h) -\n{\\mathbb F}_n(t_0) - f(t_0)n^{-{1\/3}}h]\n\\]\nand observe that\n\\[\n{\\mathbb Z}_{n,2}(h) = L_{I_n} {\\mathbb Z}_{n,2}^0(h) -\n[L_{I_n}{\\mathbb Z}_{n,2}^0(0) + (L_{I_n}{\\mathbb Z}_{n,2}^0)'(0)h\n].\n\\]\nThe unconditional convergence of ${\\mathbb Z}_{n,2}^0$ and\n$L_{I_n}{\\mathbb Z}_{n,2}^0$ follow from Corollary \\ref{cor:convdelta2}\napplied with $F_n \\equiv F$, as explained in Section \\ref{remarks}. The\nconvergence in distribution of ${\\mathbb Z}_{n,2}$ now follows from the\nContinuous Mapping theorem, using Lemma \\ref{lemma:con_derv} and\narguments similar to those in the proof of Corollary \\ref{cor:convdelta}.\n\nIt remains to show that $\\mathbb{Z}_{n,1}^*$ and $\\mathbb{Z}_{n,2}^0$\nare asymptotically independent, for example, the joint\nlimit distribution of $\\mathbb{Z}_{n,1}^*$ and $\\mathbb{Z}_{n,2}^0$ is\nthe product of their marginal limit\ndistributions. For this, it suffices to show that $(\\mathbb\n{Z}_{n,1}^*(t_1),\\ldots,\\break\\mathbb{Z}_{n,1}^*(t_k))$ and\n$(\\mathbb{Z}_{n,2}^0(s_1),\\ldots, \\mathbb{Z}_{n,2}^0(s_l))$ are\nasymptotically independent, for all choices $-\\infty< t_1 < \\cdots<\nt_k <\\infty$ and $-\\infty< s_1 < \\cdots< s_l <\\infty$. This is an\neasy consequence of\nLemma \\ref{lemma:independence} applied with $W_n^* = (\\mathbb\n{Z}_{n,1}^*(t_1), \\ldots,\\mathbb{Z}_{n,1}^*(t_k))$ and $W_n = (\\mathbb\n{Z}_{n,2}^0(s_1),\\ldots, \\mathbb{Z}_{n,2}^0(s_l))$, and $\\mathcal{F}_n\n= \\sigma(X_1,X_2,\\ldots,X_n)$.\\vspace*{1pt}\n\n(iii) We will appeal to Corollary \\ref{cor:convdelta} to find the\nunconditional distribution of $\\Delta_n^*$. We already know that\n$\\mathbb{Z}_n^*$ converges in distribution to $\\mathbb{Z}$. That (\\ref\n{eq:loc2}) holds for the limit $\\mathbb{Z}$ can be directly verified\nfrom the definition of the process. We only have to show that (\\ref\n{eq:loc}) holds unconditionally with $\\mathbb{Z}_n = \\mathbb{Z}_n^*$.\n\nLet $\\varepsilon> 0$ and $\\gamma> 0$ be given. By Proposition \\ref\n{prop:edflcm}, there exists $\\delta> 0$ and $C>0$ such that $P(A_n)\n\\ge1- \\varepsilon$ for all $n > N_0$, where\n\\[\nA_n:= \\bigl\\{ \\bigl|\\tilde F_n(t_0 + h) + \\tilde F_n(t_0) - f(t_0)h -\n\\tfrac{1}{2} f'(t_0) h^2 \\bigr| \\le\\gamma h^2 + C n^{-{2\/3}},\n\\forall|h| \\le\\delta\\bigr\\}.\n\\]\nWe can also assume that $|F(t_0 + h) + F(t_0) - f(t_0)h - (1\/2) f'(t_0)\nh^2| \\le\\gamma h^2$ for $|h| \\le\\delta$. Let ${\\mathbb Y}_n^*(h) =\nn^{2\/3}[{\\mathbb F}_n^{*}(t_0+n^{-{1\/3}}h) - {\\mathbb\nF}_n^{*}(t_0) -\nf(t_0)n^{-{1\/3}}h]$, so that ${\\mathbb Z}_n^*(h) = {\\mathbb\nY}_n^*(h) - \\Delta_n h$ for all $h \\in I_n$, and\n\\[\nL_K{\\mathbb Z}^*_n = L_K{\\mathbb Y}^*_n - \\Delta_n h\n\\]\nfor all $h \\in K$ for any interval $K \\subseteq I_n$.\n\nLet $G_n = \\tilde F_n \\mathbf{1}_{A_n} + F \\mathbf{1}_{A_n^c}$ and let\n$P_{G_n}^\\infty$ denote the\nprobability when generating the bootstrap samples from $G_n$. Then\n$G_n$ satisfies (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}), (\\ref{eq:cndtn3})\nand (\\ref{eq:cndtn4}) a.s. with $m_n = n$, $F_n = G_n$, $\\mathbb\n{F}_{n,m_n} = \\mathbb{F}^*_n \\mathbf{1}_{A_n} + \\mathbb{F}_n \\mathbf\n{1}_{A_n^c}$ and $f_n = f$. Let $J$ be a compact interval. By\nProposition \\ref{prop:loc}, applied conditionally, there exists a\ncompact interval $K$ (not depending on $\\omega$, by the \\textit{remark}\nnear the end of the proof of Proposition~\\ref{prop:loc}) such that $K\n\\supseteq J$ and\n\\[\nP_{G_n}^\\infty[L_{I_n}{\\mathbb Y}^*_n = L_{K} {\\mathbb Y}^*_n \\mbox{\non } J] (\\omega) \\ge1-\\varepsilon\n\\]\nfor $n \\ge N(\\omega) $ for a.e. $\\omega$. As $N(\\cdot)$ is bounded in\nprobability, there exists $N_1 > 0$ such that $P(B) \\ge1 - \\varepsilon$,\nwhere $B := \\{\\omega\\dvtx N(\\omega) \\le N_1\\}$. By increasing $N_1$ if\nnecessary, let us also suppose that $N_1 \\ge N_0$. Then\n\\begin{eqnarray*}\nP[L_{I_{m_n}} \\mathbb{Z}_n^* = L_K \\mathbb{Z}_n^* \\mbox{ on } J] & = &\nP[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb{Y}_n^* \\mbox{ on } J]\n\\\\\n& \\ge& \\int_{A_n} P^*[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb{Y}_n^*\n\\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& = & \\int_{A_n} P_{G_n}^\\infty[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb\n{Y}_n^* \\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& \\ge& \\int_{A_n \\cap B} P_{G_n}^\\infty[L_{I_{m_n}} \\mathbb{Y}_n^* =\nL_K \\mathbb{Y}_n^* \\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& \\ge& \\int_{A_n \\cap B} (1 - \\varepsilon) \\,dP(\\omega) \\ge1 - 3\n\\varepsilon\\qquad \\mbox{for all } n \\ge N_1\n\\end{eqnarray*}\nas $P(A_n \\cap B) \\ge1 - 2 \\varepsilon$ for $n \\ge N_1$. Thus, (\\ref\n{eq:loc}) holds and Corollary \\ref{cor:convdelta} gives $\\Delta_n^*\n\\Rightarrow(L_{\\mathbb{R}} \\mathbb{Z})'(0)$.\n\nIf (\\ref{eq:boot_limit}) holds in probability, then the unconditional\nlimit distribution of $\\Delta_n^{*}$ would be that of $2 |\\frac{1}{2}\nf(t_0) f'(t_0)|^{1\/3} \\mathbb C$, which is different from the\ndistribution of $(L_{\\mathbb R}{\\mathbb Z})'(0)$, giving rise to a\ncontradiction.\n\n(iv) We use the method of contradiction. Let $Z_n := \\mathbb\n{Z}_{n,1}^*(h_0)$ and $Y_n := \\mathbb{Z}_{n,2}(h_0)$ for some fixed\n$h_0 > 0$ (say $h_0 = 1$) and suppose that the conditional distribution\nfunction of $Z_n + Y_n = \\mathbb{Z}_n^*(h_0)$ converges in probability\nto the distribution function~$G$. By Proposition \\ref{prop:Z_conv}, the\nconditional distribution of $Z_n$ converges in probability to a normal\ndistribution, which is obviously nondegenerate. Thus, the assumptions\nof Lemma \\ref{lemma:emp_boot1} are satisfied and we conclude that $Y_n\n\\stackrel{P}{\\rightarrow} Y$, for some random variable $Y$. It then\nfollows from the Hewitt--Savage zero--one law that $Y$ is a constant, say\n$Y = c_0$ w.p. 1. The contradiction arises since $Y_n$ converges in\ndistribution to $\\mathbb{Z}_2(h_0)$ which is not a constant a.s.\n\n\\begin{figure}\n\n\\includegraphics{777f01.eps}\n\n\\caption{Scatter plot of $10\\mbox{,}000$ random draws of $((L_{\\mathbb\nR}{\\mathbb Z})'(0),(L_{\\mathbb R}{\\mathbb\nZ}_2^{0})'(0))$ when $f(t_0) = 1$ and $f'(t_0) = -2$.}\\label{fig:simul}\n\\end{figure}\n\n(v) We can show that the (unconditional) joint distribution of\n$(\\Delta_n^*, \\mathbb{Z}_{n,2}^0)$ converges to that of\n$((L_{\\mathbb{R}} {\\mathbb{Z}})'(0), {\\mathbb Z}_2^0)$. But\n$\\Delta_n^*$ and $\\mathbb{Z}_{n,2}^0$ are asymptotically independent by\nLemma \\ref{lemma:independence} applied to $W_n =\n(\\mathbb{Z}_{n,2}^0(t_1),\\mathbb{Z}_{n,2}^0(t_2),\n\\ldots,\\mathbb{Z}_{n,2}^0(t_l))$, where $t_i \\in\\mathbb{R}$, $W_n^* =\n\\Delta_n^*$ and $\\mathcal{F}_n = \\sigma(X_1,X_2,\\ldots,X_n)$. Therefore,\n$(L_{\\mathbb{R}} {\\mathbb{Z}})'(0)$ and ${\\mathbb Z}_2^0$ are\nindependent. The proposition follows directly since $\\mathbb{Z}_2$ is a\nmeasurable function of $\\mathbb{Z}_{2}^0$.\n\\end{pf}\n\nIf the conditional distribution of $\\Delta^*_n$ converges in\nprobability, as a consequence of (v) of Theorem\n\\ref{thm:bootmle}, $(L_{\\mathbb R}{\\mathbb Z})'(0)$ and $(L_{\\mathbb\nR}{\\mathbb Z}_2^{0})'(0)$ must also be independent. Figure \\ref\n{fig:simul} shows the scatter plot of $(L_{\\mathbb R}{\\mathbb Z})'(0)$\nand $(L_{\\mathbb R}{\\mathbb Z}_2^{0})'(0)$ obtained from a simulation\nstudy with $10\\mbox{,}000$ samples, $f(t_0) = 1$ and $f'(t_0) = -2$. The\ncorrelation coefficient obtained $-0.2999$ is highly significant\n($p$-value $< 0.0001$). Thus, when combined with simulations, (v) of\nTheorem \\ref{thm:bootmle} strongly suggests that the conditional\ndistribution of $\\Delta_n^*$ does not converge in probability.\n\n\\subsection{Bootstrapping from the EDF}\n\nA similar, slightly simpler\npattern arises if the bootstrap sample is drawn\nfrom $\\hat{F}_n = {\\mathbb F}_n$. Define ${\\mathbb Z}_n^*$ as before,\nand let $\\mathbb{Z}_{n,1}^* (h) = n^{2\/3} \\{\n(\\mathbb{F}_n^*-{\\mathbb F}_n)(t_0 + n^{-{1\/3}}h) - (\\mathbb\n{F}_n^*- {\\mathbb F}_n)(t_0)\\}$ and\n$\\mathbb{Z}_{n,2}(h) = n^{2\/3} \\{{\\mathbb F}_n(t_0 + h n^{-{1\/3}}) -\n{\\mathbb F}_n(t_0) - \\tilde f_n(t_0)\nn^{-{1\/3}}h\\}$. Then ${\\mathbb Z}_n^* = {\\mathbb Z}_{n,1}^* +\n{\\mathbb Z}_{n,2}$. Recall the definition of the\nprocesses $\\mathbb{W}_1$, $\\mathbb{W}_2$, $\\mathbb{Z}_1$, $\\mathbb\n{Z}^0_2$ in Section \\ref{bootsNPMLE}. Define\n\\[\n\\mathbb{Z}_2(h) = \\mathbb{Z}_2^0(h) -(L_{\\mathbb{R}} \\mathbb\n{Z}_2^0)'(0) h.\n\\]\n\\begin{theorem}\\label{thm:bootmle2}\n\\textup{(i)} The conditional distribution of $\\mathbb{Z}_{n,1}^*$ given\n$\\mathbf{X} = (X_1,X_2,\\break\\ldots)$ converges a.s. to the distribution\nof $\\mathbb{Z}_1$.\n\n{\\smallskipamount=0pt\n\\begin{longlist}[(iii)]\n\\item[(ii)] The unconditional distribution of $\\mathbb{Z}_{n,2}$\nconverges to that of $\\mathbb{Z}_2$ and the\nunconditional distributions of $(\\mathbb{Z}_{n,1}^{*},\\mathbb\n{Z}_{n,2})$, and $\\mathbb{Z}_n^*$ converge to those of $(\\mathbb\n{Z}_{1},\\mathbb{Z}_{2})$ and~$\\mathbb{Z}$.\n\n\\item[(iii)] The unconditional distribution of $\\Delta_n^*$ converges\nto that of $(L_{\\mathbb{R}}\\mathbb{Z})'(0)$, and (\\ref{eq:boot_limit}) fails.\n\n\\item[(iv)] Conditional on $\\mathbf{X}$, the distribution of $\\mathbb\n{Z}_n^*$ does not have a weak limit in\nprobability.\n\n\\item[(v)] If the conditional distribution function of $\\Delta_n^*$\nconverges in probability, then\n$(L_{\\mathbb{R}}{\\mathbb{Z}})'(0)$ and ${\\mathbb Z}_2$ must be independent.\n\\end{longlist}}\n\\end{theorem}\n\\begin{Remark*} The proof of this theorem runs along similar lines to\nthat of Theorem \\ref{thm:bootmle}. We briefly\nhighlight the differences.\n\\end{Remark*}\n\n\\begin{longlist}\n\\item The conditional convergence of $\\mathbb{Z}_{n,1}^*$ follows\nfrom Proposition \\ref{prop:Z_conv} with $m_n = n$, $F_n = \\mathbb\n{F}_n$, $\\mathbb{F}_{n,m_n} = \\mathbb{F}_n^*$, applied conditionally\ngiven ${\\mathbf X}$. It is only necessary to show that (\\ref\n{eq:cndtn1}) is satisfied almost surely, and this follows from the Law\nof the Iterated Logarithm for ${\\mathbb F}_n$, as explained in\nSection \\ref{remarks}. Then the unconditional limiting distribution of\n$\\mathbb\n{Z}_{n,1}^*$ must also be that of $\\mathbb{Z}_1$.\n\n\\item The proof is similar to that of (ii) of Theorem \\ref\n{thm:bootmle}, except that now $\\mathbb{Z}_{n,2}(h) = \\mathbb\n{Z}^0_{n,2}(h) - (L_{I_n} \\mathbb{Z}^0_{n,2})'(0) h$.\n\\end{longlist}\n\nThe proofs of (iii)--(v) are very similar to that of (iii)--(v) of\nTheorem \\ref{thm:bootmle}.\n\n\\subsection{Performance of the bootstrap methods in finite\nsamples}\\label{simul}\n\nIn this subsection, we illustrate the poor finite sample performance\nof the two inconsistent bootstrap schemes, namely, bootstrapping from\nthe EDF $\\mathbb{F}_n$ and the NPMLE $\\tilde F_n$. Table~\\ref\n{PerfBootsMeth} shows the estimated coverage probabilities of nominal\n95\\% confidence intervals for $f(1)$ using the two bootstrap methods\n\\begin{table}[b]\n\\caption{Estimated coverage probabilities of nominal 95\\% confidence\nintervals for $f(1)$ while bootstrapping from the EDF $\\mathbb{F}_n$\nand NPMLE $\\tilde F_n$, with varying sample size $n$ for the two\nmodels: $\\operatorname{Exponential}(1)$ (left) and $|Z|$ where $Z \\sim\n\\operatorname{Normal}(0,1)$ (right)}\\label{PerfBootsMeth}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}lccccc@{}}\n\\hline\n$\\bolds n$ & \\textbf{EDF} & \\textbf{NPMLE} & $\\bolds n$ & \\textbf{EDF} & \\textbf{NPMLE} \\\\\n\\hline\n\\phantom{0}$50$ & 0.747 & 0.720 & \\phantom{0}$50$ & 0.761 & 0.739 \\\\\n$100$ & 0.776 & 0.755  &  $100$ & 0.778 & 0.757 \\\\\n$200$ & 0.802 & 0.780  &  $200$ & 0.780 & 0.762 \\\\\n$500$ & 0.832 & 0.797  &  $500$ & 0.788 & 0.755 \\\\\n\\hline\n\\end{tabular*}\n\\end{table}\nfor different sample sizes, when the true distribution is assumed to be\nExponential(1) and $|\\mathrm{Normal}(0,1)|$, respectively. We used 1000\nbootstrap samples to compute each confidence interval and then\nconstructed 1000 such confidence intervals to estimate the actual\ncoverage probabilities. As is clear from the table the coverage\nprobabilities fall well short of the nominal 0.95 value. Leger and\nMacGibbon (\\citeyear{legerMa06}) also illustrate such a discrepancy in\nthe nominal and\nactual coverage probabilities while bootstrapping from the EDF for the\nChernoff's estimator of the mode.\n\n\nFigure \\ref{fig:InConsEDF_Hist} shows the histograms (computed from\n10,000 bootstrap samples) of the two inconsistent bootstrap\ndistributions obtained from a single sample of 500 Exponential(1)\nrandom variables along with the histogram of the exact distribution of\n$\\Delta_n$ (obtained from simulation). The bootstrap distributions are\nskewed and have very different shapes and supports compared to that on\nthe left panel of Figure \\ref{fig:InConsEDF_Hist}. The histograms\nillustrate the inconsistency of the bootstrap procedures.\n\n\\begin{figure}\n\n\\includegraphics{777f02.eps}\n\n\\caption{Histograms of the exact distribution of $\\Delta_n$ (left\npanel) and the two bootstrap distributions while drawing bootstrap\nsamples from $\\mathbb{F}_n$ (middle panel) and $\\tilde F_n$ (right\npanel) for $n = 500$.}\\label{fig:InConsEDF_Hist}\n\\end{figure}\n\n\\begin{figure}[b]\n\n\\includegraphics{777f03.eps}\n\n\\caption{Estimated 0.95 quantile of the bootstrap distribution while\ngenerating the bootstrap samples from $\\mathbb{F}_n$ (dashed lines) and\n$\\tilde F_n$ (solid-dotted lines) for two independent data sequences\nalong with the 0.95 quantile of the limit distribution of $\\Delta_n$\n(solid line) for the two models: $\\operatorname{Exponential}(1)$ (left panel) and $|Z|$\nwhere $Z \\sim\\operatorname{Normal}(0,1)$ (right panel).}\\label{fig:InConsQuantile}\n\\end{figure}\n\nThe estimated coverage probabilities in Table \\ref{PerfBootsMeth} are\nunconditional [see (iii) of Theorems \\ref{thm:bootmle} and \\ref\n{thm:bootmle2}] and do not provide direct evidence to suggest that the\nconditional distribution of $\\Delta_n^*$ does not converge in\nprobability. Figure \\ref{fig:InConsQuantile} shows the estimated 0.95\nquantile of the bootstrap distribution for two independent data\nsequences as the sample size increases from 500 to 10,000, for the two\nbootstrap procedures, and for both the models (exponential and normal).\nThe bootstrap quantile fluctuates enormously even at very large sample\nsizes and shows signs of nonconvergence. If the bootstrap were\nconsistent, the estimated quantiles should converge to 0.6887 (0.8269),\nthe 0.95 quantile of the limit distribution of $\\Delta_n$, indicated by\nthe solid line in Figure \\ref{fig:InConsQuantile}. From the left panel\nof Figure \\ref{fig:InConsQuantile}, we see that the estimated bootstrap\n0.95 quantiles (obtained from the two procedures) for one data sequence\nstays below 0.6887, while for the other, the 0.95 quantiles stay above\n0.6887, indicating the strong dependence on the sample path. Note that\nif the bootstrap distributions had a limit, then Figure \\ref\n{fig:InConsQuantile} suggests that the limit varies with the sample\npath, and that is impossible as explained in Section \\ref\n{generalities}. This provides evidence for the nonconvergence of the\nbootstrap estimator.\n\n\n\\section{Consistent bootstrap methods}\\label{smooth_boots}\n\nThe main reason for the inconsistency of bootstrap methods discussed in\nthe previous section is the lack of smoothness of the distribution\nfunction from which the bootstrap samples are generated. The EDF\n$\\mathbb{F}_n$ does not have a density, and $\\tilde F_n$ does not have\na differentiable density, whereas $F$ is assumed to have a nonzero\ndifferentiable density at $t_0$. At a more technical level, the lack of\nsmoothness manifests itself through the failure of (\\ref{eq:cndtn2}).\n\nThe results from Section \\ref{prelim} may be directly applied to derive\nsufficient conditions on the smoothness of the distribution from which\nthe bootstrap samples are generated. Let\n$X_1,X_2,\\ldots\\stackrel{\\mathrm{ind}}{\\sim}\nF$; let $\\hat F_n$ be an estimate of $F$ computed from $X_1,\\ldots\n,X_n$; and let $\\hat f_n$ be the density of $\\hat F_n$ or a surrogate,\nas in Section \\ref{boots_prob}.\n\\begin{theorem}\\label{thm:cons_sm_boot} If (\\ref{eq:cndtn0}), (\\ref\n{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) hold a.s. with $F_n = \\hat F_n$ and $f_n = \\hat f_n$,\nthen the bootstrap estimate is strongly\nconsistent, for example, (\\ref{eq:boot_limit}) holds w.p. 1. In\nparticular, the bootstrap estimate is strongly consistent if\nthere is a $\\delta> 0$ for which $\\hat F_n$ has a continuously\ndifferentiable density $\\hat f_n$ on $[t_0 - \\delta,t_0 + \\delta]$, and\n(\\ref{eq:cndsmoothF}) holds a.s. with $F_n = \\hat F_n$ and $f_n = \\hat f_n$.\n\\end{theorem}\n\\begin{pf} That $\\Delta_n^*$ converges weakly to the distribution on\nthe right-hand side of (\\ref{eq:chrnff}) a.s. follows from Corollary\n\\ref\n{cor:convdelta2} applied conditionally given $\\mathbf{X}$ with $F_n =\n\\hat F_n$ and $f_n = \\hat f_n$. The second assertion follows similarly\nfrom Corollary \\ref{cor:smoothF}.\n\\end{pf}\n\n\\subsection{Smoothing $\\tilde F_n$}\n\nWe show that generating bootstrap\nsamples from a suitably smoothed version of\n$\\tilde{F}_n$ leads to a consistent bootstrap procedure. To avoid\nboundary effects and ensure that the smoothed version has a decreasing\ndensity on $(0,\\infty)$, we use a logarithmic transformation. Let $K$\nbe a twice continuously differentiable symmetric density for which\n\\begin{equation}\n\\label{eq:kay}\n\\int_{-\\infty}^{\\infty} [K(z)+|K'(z)|+|K''(z)|]e^{\\eta|z|}\\,dz < \\infty\n\\end{equation}\nfor some $\\eta> 0$. Let\n\\begin{eqnarray}\\label{eq:fchck1}\nK_h(x,u) & = & {1\\over hx}K \\biggl[{1\\over h}\\log\\biggl({u\\over x}\\biggr) \\biggr]\\quad\n\\mbox{and} \\nonumber\\\\[-8pt]\\\\[-8pt]\n\\check{f}_n(x) & = & \\int_{0}^{\\infty} K_h(x,u)\\tilde{f}_n(u)\\,du = \\int\n_{0}^{\\infty} K_h(1,u)\\tilde{f}_n(xu)\\,du.\\nonumber\n\\end{eqnarray}\nThus, $e^y\\check{f}_n(e^y) = \\int_{-\\infty}^{\\infty}\nh^{-1}K[h^{-1}(y-z)] \\tilde{f}_n(e^z)e^z \\,dz$. Integrating and using\ncapital letters to denote distribution functions,\n\\begin{eqnarray*}\n\\label{eq:fchck2}\n\\check{F}_n(e^y) & = & \\int_{-\\infty}^y \\check f_n(e^s) e^s \\,ds\n\\\\\n& = & \\int_{-\\infty}^y \\int_{-\\infty}^\\infty\\frac{1}{h} K \\biggl( \\frac\n{s - v}{h} \\biggr) \\tilde f_n(e^v) e^v \\,dv \\,ds\n\\\\\n& = & \\int_{-\\infty}^{\\infty} K(z)\\tilde{F}_n(e^{y-hz})\\,dz.\n\\end{eqnarray*}\nAlternatively, integrating (\\ref{eq:fchck1}) by parts yields\n\\[\n\\check{f}_n(x) = -\\int_{0}^{\\infty} {\n\\partial\\over\n\\partial u}K_h(x,u)\\tilde{F}_n(u)\\,du.\n\\]\nThe proof of (\\ref{eq:boot_limit}) requires showing that $\\check{F}_n$\nand its derivatives are sufficiently close to\nthose of $F$, and it is convenient to separate the estimation error\n$\\check{F}_n-F$ into sampling and approximation\nerror. Thus, let\n\\begin{equation}\n\\label{eq:fbar}\n\\bar{F}_h(e^y) = \\int_{-\\infty}^{\\infty} K(z) F(e^{y-hz})\\,dz.\n\\end{equation}\nWe denote the first and second derivatives of $\\bar{F}_h$ by $\\bar\n{f}_h$ and $\\bar{f}_h'$, respectively. Recall that $F$ is assumed to\nhave a nonincreasing density on $(0,\\infty)$ that is continuously\ndifferentiable near $t_0$.\n\\begin{lemma}\\label{lem:fbar} ${\\lim_{h \\rightarrow0}} \\| \\bar F_h - F\\|\n= 0$, and there is a $\\delta> 0$ for which\n\\begin{equation}\n\\label{eq:fbar2}\n\\lim_{h\\to0} \\sup_{|x-t_0|\\le\\delta} [ |\\bar{f}_h(x)-f(x)| +\n|\\bar{f}_h'(x)-f'(x)| ] = 0.\n\\end{equation}\n\\end{lemma}\n\\begin{pf} First, observe that\n\\[\n\\bar{F}_h(e^y)-F(e^y) = \\int_{-\\infty}^{\\infty}\nK(z)[F(e^{y-hz})-F(e^y)]\\,dz\n\\]\nby (\\ref{eq:fbar}). That $\\lim_{h \\rightarrow0} \\bar F_h(x) = F(x)$\nfor all $x \\ge0$ follows easily from the Dominated Convergence\ntheorem, and uniform convergence then follows from Polya's theorem.\nThis establishes the first assertion of the lemma. Next, consider (\\ref\n{eq:fbar2}). Given $t_0 > 0$, let $y_0 = \\log(t_0)$ and let $\\delta>\n0$ be so small that $e^yf(e^y)$ is continuously differentiable (in $y$)\non $[y_0-2\\delta,y_0+2\\delta]$. Then\n\\begin{eqnarray*}\n\\bar{f}_h(x) - f(x) & = & \\int_{-\\infty}^{\\infty} K(z)[f(xe^{hz})-f(x)]\ne^{h z} \\,dz \\\\\n&&{} + f(x) \\int_{-\\infty}^{\\infty} (e^{hz} - 1) K(z) \\,dz\n\\end{eqnarray*}\nand thus\n\\[\n{\\sup_{|x-t_0|\\le\\delta}} |\\bar{f}_h(x)-f(x)| \\le{\\int_{-\\infty}^{\\infty\n} \\sup_{|x-t_0|\\le\\delta}} |f(xe^{hz}) - f(x)|\ne^{h z} K(z)\\,dz + O(h^2)\n\\]\nfor any $0 < \\delta< t_0$. For sufficiently small $\\delta$, the\nintegrand approach zero as $h \\to0$; and it is bounded by $\\sup\n_{|x-t_0|\\le\\delta} (e^{-hz}\/x + f(x)) e^{hz} K(z)$, since $f(x) \\le\n1\/x$ for all $x > 0$. So the right-hand side approaches zero as $h \\to\n0$ by\nthe Dominated Convergence theorem. That ${\\sup_{|x-t_0|\\le\\delta}} |\\bar\n{f}_h'(x)-f'(x)| \\to0$ may be established similarly.\n\\end{pf}\n\\begin{theorem} Let $K$ be a twice continuously differentiable,\nsymmetric density for which (\\ref{eq:kay}) holds. If\n\\[\nh = h_n \\to0 \\quad\\mbox{and}\\quad h_n^{2}\\sqrt{n\\over\\log\\log(n)}\n\\to\\infty,\n\\]\nthen the bootstrap estimator is strongly consistent; that is, (\\ref\n{eq:boot_limit}) holds a.s.\n\\end{theorem}\n\\begin{pf} By Theorem\\vspace*{1pt} \\ref{thm:cons_sm_boot}, it suffices to show that\n(\\ref{eq:cndsmoothF}) holds a.s. with $\\hat\nF_n = \\check F_n$ and $\\hat f_n = \\check f_n$; and this would follow\nfrom\n\\[\n\\|\\check F_n - \\bar F_h\\| + \\sup_{|x - t_0| \\le\\delta} [|\\check f_n(x)\n- \\bar f_h(x)| + |\\check f_n'(x) - \\bar\nf_h'(x)|] \\rightarrow0 \\qquad\\mbox{a.s.}\n\\]\nfor some $\\delta> 0$ and Lemma \\ref{lem:fbar}. Clearly, using (\\ref\n{eq:fchck2}),\n\\begin{equation}\\label{eq:Fn-Fh}\n\\check{F}_n(e^y)-\\bar{F}_h(e^y) = \\frac{1}{h} \\int_{-\\infty}^{\\infty}\n[\\tilde{F}_n(e^{t}) - F(e^{t})] K \\biggl( \\frac{y\n- t}{h} \\biggr) \\,dt\n\\end{equation}\nfor all $y$, so that\n\\[\n\\| \\check{F}_n - \\bar F_h \\| \\le\\|{\\tilde F}_n - F \\| \\le\\|{\\mathbb\nF}_n - F \\| = O \\bigl[ \\sqrt{\\log\\log(n)\/n}\n\\bigr] \\qquad\\mbox{a.s.}\n\\]\nby Marshall's lemma and the Law of the Iterated Logarithm.\nDifferentiating (\\ref{eq:Fn-Fh}) gives\n\\[\n\\check f_n(e^{y}) - \\bar f_h(e^{y}) = \\frac{e^{-y}}{h^2} \\int_{-\\infty\n}^{\\infty} [\\tilde{F}_n(e^{t}) - F(e^{t})]\nK' \\biggl( \\frac{y - t}{h} \\biggr) \\,dt.\n\\]\nDifferentiating (\\ref{eq:Fn-Fh}) again and then taking absolute values\nand considering $0 < h \\le1$, we get\n\\begin{eqnarray*}\n&&\\hspace*{-6pt} \\sup_{|x - t_0| \\le\\delta} \\{|\\check f_n(x) - \\bar f_h(x)| +\n|\\check f_n'(x) - \\bar f_h'(x)| \\} \\\\\n&&\\hspace*{-6pt}\\qquad \\le \\frac{M}{h^3} \\sup_{|x - t_0| \\le\\delta} \\int_{-\\infty\n}^{\\infty} |\\tilde{F}_n(e^{t}) - F(e^{t})| \\biggl[ \\biggl| K' \\biggl( \\frac\n{\\log x -\nt}{h} \\biggr) \\biggr| + \\biggl| K'' \\biggl( \\frac{\\log x - t}{h} \\biggr)\n\\biggr| \\biggr] \\,dt \\\\\n&&\\hspace*{-6pt}\\qquad \\le \\frac{M}{h^2} \\|{\\mathbb F}_n - F \\| \\int_{-\\infty}^{\\infty}\n[|K'(z)| + |K''(z)|] \\,dz\n\\rightarrow 0 \\qquad\\mbox{a.s.}\n\\end{eqnarray*}\nfor a constant $M > 0$, as $h_n^2 \\sqrt{n\/ \\log\\log(n)} \\rightarrow\n\\infty$, where Marshall's lemma and the Law of Iterated Logarithm have\nbeen used again.\n\\end{pf}\n\n\\subsection{$m$ out of $n$ bootstrap}\\label{m_n_boots}\n\nIn Section \\ref\n{boots_prob}, we showed that the two most intuitive methods of\nbootstrapping are inconsistent. In this section, we show that the\ncorresponding $m$ out of $n$ bootstrap procedures are weakly consistent.\n\\begin{theorem}\\label{thm:m_n_boot1} If $\\hat F_n = \\mathbb{F}_n$,\n$\\hat f_n = \\tilde f_n$, and $m_n = o(n)$ then the\nbootstrap procedure is weakly consistent, for example, (\\ref\n{eq:boot_limit}) holds in probability.\n\\end{theorem}\n\\begin{pf}\nConditions (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4})\nhold a.s. from (\\ref{eq:lil}), as\nexplained in Section \\ref{remarks}. To verify (\\ref{eq:cndtn3}), let\n$\\gamma> 0$ be given. From the proof of Proposition \\ref{prop:loc}\n[also see Kim and Pollard (\\citeyear{kimPo90}), page 218], there exists\n$\\delta>0$\nsuch that $|\\mathbb{F}_{n}(t_0 + h) - \\mathbb{F}_{n}(t_0) - F(t_0 + h)\n- F(t_0)| \\le\\gamma h^2 + \\mathcal{C}_n n^{-2\/3}$,\nfor $|h| \\le\\delta$, where $\\mathcal{C}_n$'s are random variables of\norder $O_P(1)$. We can also assume that $|F(t_0 + h) + F(t_0) - f(t_0)h\n- (1\/2) f'(t_0) h^2| \\le(1\/2) \\gamma h^2$ for $|h| \\le\\delta$. Then,\nusing the inequality $2|a b| \\le\\gamma a^2 + b^2\/\\gamma$,\n\\begin{eqnarray}\\label{eq:A_nB_nC_n}\\qquad\n&& \\biggl| \\mathbb{F}_{n}(t_0 + h) - \\mathbb{F}_{n}(t_0)\n- h \\tilde f_n(t_0) - \\frac{1}{2}\nh^2 f'(t_0) \\biggr| \\nonumber\\\\\n&&\\qquad \\le \\biggl| \\mathbb{F}_{n}(t_0 + h) - \\mathbb\n{F}_{n}(t_0) - h f(t_0) - \\frac{1}{2}\nh^2 f'(t_0) \\biggr| + |h| |\\tilde f_n(t_0) - f(t_0) |\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad \\le \\biggl\\{\\gamma h^2 + \\mathcal{C}_n n^{-{2 \/\n3}} + \\frac{1}{2} \\gamma h^2 \\biggr\\} + \\biggl\\{ \\frac{1}{2} \\gamma h^2\n+ \\frac{1}{2 \\gamma} |\\tilde\nf_n(t_0) - f(t_0) |^2 \\biggr\\}\\nonumber\\\\\n&&\\qquad \\le 2 \\gamma h^2 + \\mathcal{C}_n n^{-{2 \/3}} +\nO_P(n^{-2\/3}) \\le2 \\gamma h^2 + o_P(m_n^{-{2 \/3}}).\\nonumber\n\\end{eqnarray}\nFor (\\ref{eq:cndtn2}), write\n\\begin{eqnarray}\n\\label{eq:m_n_boot_t2}\n&& m_n^{{2\/3}} \\{\\mathbb{F}_n(t_0 + m_n^{-{1\/3}}h) - \\mathbb\n{F}_n(t_0) - m_n^{-{1\/3}}\\tilde\nf_n(t_0)h\\} \\nonumber\\\\\n&&\\qquad = m_n^{{2\/3}} \\{ (\\mathbb{F}_n - F)(t_0 + m_n^{-{1\/3}}h) -\n(\\mathbb{F}_n - F)(t_0) \\}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\quad{} + m_n^{{1\/3}} [f(t_0)-\\tilde{f}_n(t_0)]h + \\tfrac{1}{\n2}f'(t_0)h^2 + o(1) \\nonumber\\\\\n&&\\qquad \\stackrel{P}{\\rightarrow} \\tfrac{1}{2}f'(t_0)h^2\\nonumber\n\\end{eqnarray}\nuniformly on compacts using Hungarian Embedding to bound the second\nline and (\\ref{eq:chrnff}) (and a two-term Taylor\nexpansion) in the third.\n\nGiven any subsequence $\\{n_k\\} \\subset\\mathbb{N}$, there exists a\nfurther subsequence $\\{n_{k_l}\\}$ such that\n(\\ref{eq:A_nB_nC_n}) and (\\ref{eq:m_n_boot_t2}) hold a.s. and\nTheorem \\ref{thm:cons_sm_boot} is applicable. Thus,\n(\\ref{eq:boot_limit}) holds for the subsequence $\\{n_{k_l}\\}$, thereby\nshowing that (\\ref{eq:boot_limit}) holds in\nprobability.\n\\end{pf}\n\nNext consider bootstrapping from $\\tilde{F}_n$. We will assume slightly\nstronger conditions on $F$, namely, conditions (a)--(d) mentioned in\nTheorem 7.2.3 of Robertson, Wright and Dykstra (\\citeyear{RWD88}):\n\\begin{itemize}\n\\item[(a)] $\\alpha_1(F) = \\inf\\{x\\dvtx F(x) = 1\\} < \\infty$,\n\n\\item[(b)] $F$ is twice continuously differentiable on $(0,\\alpha_1(F))$,\n\n\\item[(c)] $\\gamma(F) = \\frac{{\\sup_{0 < x < \\alpha_1(F)}} |f'(x)|}{\\inf\n_{0 < x < \\alpha_1(F)} f^2(x)} < \\infty$,\n\n\\item[(d)] $\\beta(F) = {\\inf_{0 < x < \\alpha_1(F)}} |\\frac\n{-f'(x)}{f^2(x)}| > 0$.\n\\end{itemize}\n\\begin{theorem}\\label{thm:m_n_boot2} Suppose that \\textup{(a)--(d)} hold. If\n$\\hat{F}_n = \\tilde{F}_n$, $\\hat f_n = \\tilde f_n$, and $m_n = o[n\n(\\log n)^{-{3\/2}}]$ then (\\ref{eq:boot_limit}) holds in probability.\n\\end{theorem}\n\\begin{pf} Conditions (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref\n{eq:cndtn4}) again follow from (\\ref{eq:lil}), as explained in\nSection \\ref{remarks}. The verification of (\\ref{eq:cndtn3}) is\nsimilar to the argument in the proof of Theorem \\ref{thm:m_n_boot1}. We\nshow that (\\ref{eq:cndtn2}) holds. Adding and subtracting $m_n^{{2\/\n3}} [ \\mathbb{F}_n(t_0 + m_n^{-{1\/3}} h) - \\mathbb{F}_n(t_0)]$\nfrom $\\mathbb{Z}_{n,2}(h)$ and using\n(\\ref{eq:m_n_boot_t2}) and the result of Kiefer and Wolfowitz (\\citeyear{KW76})\n\\begin{eqnarray*}\n\\sup_{|h| \\le c} \\biggl| {\\mathbb Z}_{n,2}(h) - {1\\over2}f'(t_0)h^2\n\\biggr| & \\le& 2 m_n^{2\/3} \\| \\tilde{F}_n - {\\mathbb F}_n\\|\n+ o_P(1) \\\\\n& \\le& 2 m_n^{2 \/3} \\| \\tilde{F}_n - {\\mathbb F}_n\\| + o_P(1)\n\\\\\n& = & O_P[m_n^{2\/3} n^{-{2\/3}}\\log(n)] + o_P(1)\n\\end{eqnarray*}\nfor any $c > 0$ from which (\\ref{eq:cndtn2}) follows easily.\n\\end{pf}\n\n\\section{Discussion}\\label{discussion}\n\nWe have shown that bootstrap estimators are inconsistent when bootstrap\nsamples are drawn from either the EDF ${\\mathbb F}_n$ or its least\nconcave majorant ${\\tilde F}_n$ but consistent when the bootstrap\nsamples are drawn from a smoothed version of $\\tilde{F}_n$ or an $m$\nout of $n$ bootstrap is used. We have also derived necessary conditions\nfor the bootstrap estimator to have a conditional weak limit, when\nbootstrapping from either ${\\mathbb F}_n$ or ${\\tilde F}_n$ and\npresented compelling numerical evidence that these conditions are not\nsatisfied. While these results have been obtained for the Grenander\nestimator, our results and findings have broader implications for the\n(in)-consistency of the bootstrap methods in problems with an $n^{1\/3}$\nconvergence rate.\n\nTo illustrate the broader implications, we contrast our finding with\nthose of Abrevaya and Huang (\\citeyear{AH05}), who considered a more\ngeneral framework, as in Kim and Pollard (\\citeyear{kimPo90}). For\nsimplicity, we use the same notation as in Abrevaya and Huang\n(\\citeyear{AH05}). Let $W_n := r_n (\\theta _n - \\theta_0)$ and $\\hat\nW_n := r_n (\\hat\\theta_n - \\theta_n)$ be the sample and bootstrap\nstatistics of interest. In our case $r_n = n^{1\/3}$, $\\theta_0 =\nf(t_0)$, $\\theta_n = \\tilde f_n(t_0)$ and $\\hat \\theta_n = \\tilde\nf_n^*(t_0)$. When specialized to the Grenander estimator, Theorem $2$\nof Abrevaya and Huang (\\citeyear{AH05}) would imply [by calculations\nsimilar to those in their Theorem~5 for the NPMLE in a binary choice\nmodel] that\n\\[\n\\hat W_n \\Rightarrow\\arg\\max\\hat Z(t) - \\arg\\max Z(t)\n\\]\nconditional on the original sample, in $P^\\infty$-probability, where\n$Z(t) = W(t) - ct^2$ and $\\hat Z(t) = W(t) + \\hat W(t) - ct^2$, $W$ and\n$\\hat W$ are two independent two sided Brownian motions on $\\mathbb{R}$\nwith $W(0) = \\hat W(0) = 0$ and $c$ is a positive constant depending on\n$F$. We also know that $W_n \\Rightarrow\\arg\\max Z(t)$\nunconditionally. By (v) of Theorem \\ref{thm:bootmle}, this would\nforce the independence of $\\arg\\max Z(t)$ and $\\arg\\max\\hat Z(t) -\n\\arg\\max Z(t)$; but, there is overwhelming numerical evidence that\nthese random variables are correlated.\n\n\\begin{appendix}\\label{app}\n\\section*{Appendix}\n\n\\begin{lemma}\\label{lemma:slope_lcm} Let $\\Psi\\dvtx\\mathbb{R}\n\\rightarrow\n\\mathbb{R}$ be a function such that $\\Psi(h) \\le M$ for all $h \\in\n\\mathbb{R}$, for some $M > 0$, and\n\\begin{equation}\n\\label{eq:Psi_prop} \\lim_{|h| \\rightarrow\n\\infty} \\frac{\\Psi(h)}{|h|} = -\\infty.\n\\end{equation}\nThen for any $b > 0$, there exists $c_0 > b$ such that for any $c \\ge\nc_0$, $L_{\\mathbb{R}} \\Psi(h) = L_{[-c,c]} \\Psi(h)$ for all $|h| \\le b$.\n\\end{lemma}\n\\begin{pf} Note that for any $c > 0$, $L_{\\mathbb{R}} \\Psi(h)\n\\ge L_{[-c,c]} \\Psi(h)$ for all $h \\in[-c,c]$. Given $b > 0$,\nconsider $c > b$ and $\\Phi_c(h) = L_{[-c,c]} \\Psi(h)$ for $h \\in\n[-b,b]$, and let $\\Phi_c$ be the linear extension of $L_{[-c,c]} \\Psi\n|_{[-b,b]}$ outside $[-b,b]$. We will show that there\nexists $c_0 > b + 1$ such that $\\Phi_{c_0} \\ge\\Psi$. Then $\\Phi_{c_0}$\nwill be a concave function everywhere greater than $\\Psi$, and thus\n$\\Phi_{c_0} \\ge L_{\\mathbb{R}} \\Psi$. Hence, $L_{\\mathbb{R}} \\Psi(h)\n\\le\\Phi_{c_0}(h) = L_{[-c_0,c_0]} \\Psi(h)$ for $h \\in[-b,b]$,\nyielding the desired result.\n\nFor any $c > b + 1$, $\\Phi_c(h) = \\Phi_c(b) - \\Phi_c'(b) + \\Phi_c'(b)\n(h - b + 1)$ for $h \\ge b$. Using the min--max formula,\n\\begin{eqnarray*}\n\\Phi_c'(b) & = & \\min_{-c \\le s \\le b} \\max_{b \\le t \\le c}\n\\frac{\\Psi(t) - \\Psi(s)}{t - s} \\\\\n& \\ge& \\min_{-c \\le s \\le b} \\frac{\\Psi( b + 1) - \\Psi(s)}{ (b + 1) -\ns} \\\\\n&\\ge& \\Psi(b + 1) - M =: B_0 \\le0.\n\\end{eqnarray*}\nThus,\n\\begin{eqnarray*}\n\\Phi_c(h) & = & \\Phi_c(b) - \\Phi_c'(b) + \\Phi_c'(b) (h - b + 1)\n\\\\\n& \\ge& \\{\\Psi(b) - \\Phi_c'(b)\\} +\n\\Phi_c'(b) (h - b + 1) \\\\\n& \\ge& \\Psi(b) + (h - b) B_0\n\\end{eqnarray*}\nfor $h \\ge b + 1$. Observe that $B_0$ does not depend on $c$. Combining\nthis with a similar calculation for $h < -(b + 1)$, there are $K_0 \\ge\n0$ and $K_1 \\ge0$, depending only on $b$, for which $\\Phi_c(h) \\ge K_0\n- K_1 |h|$ for $|h| \\ge b + 1$. From (\\ref{eq:Psi_prop}), there is $c_0\n> b + 1$ for which $\\Psi(h) \\le K_0 - K_1 |h|$ for all $|h| \\ge c_0$ in\nwhich case $\\Psi(h) \\le\\Phi_{c_0} (h)$ for all $h$. It follows that\n$L_{\\mathbb R} \\Psi\\le\\Phi_{c_0}(h)$ for $|h| \\le b$.\n\\end{pf}\n\\begin{lemma}\\label{lem:abc}\nLet ${\\mathbb B}$ be a standard Brownian motion. If $a, b, c > 0,\na^3b = 1$, then\n\\begin{equation}\\label{eq:abc1}\nP \\biggl[\\sup_{t \\in\\mathbb{R}} {|{\\mathbb B}(t)|\\over a+bt^2} > c\n\\biggr] = P\\biggl[ \\sup_{s \\in\\mathbb{R}} {|{\\mathbb B}(s)|\\over1+s^2}\n> c \\biggr].\n\\end{equation}\n\\end{lemma}\n\\begin{pf} This follows directly from rescaling properties of Brownian\nmotion by letting $t = a^2s$.\n\\end{pf}\n\\begin{pf*}{Proof of Proposition \\ref{prop:loc}}\nLet $J = [a_1,a_2]$ and $\\varepsilon> 0$ be as in the statement of the\nproposition; let $\\gamma= |f'(t_0)|\/16$; and recall (\\ref\n{eq:kmt}) and (\\ref{eq:boots_proc1}) from the proof of Proposition \\ref\n{prop:Z_conv}. Then there exists $0 < \\delta< 1$, $C \\ge1$, and $n_0\n\\ge1$ for which (\\ref{eq:cndtn3}) and (\\ref{eq:cndtn4}) hold for all\n$n \\ge n_0$. Let $I_{m_n}^* := [-\\delta m_n^{1\/3}, \\delta\nm_n^{1\/3}]$. By making $\\delta$ smaller,\\vspace*{-1pt} if necessary, and using\nLemma \\ref{lem:loc}, $L_{I_{m_n}}{\\mathbb Z}_n(h) =\nL_{I_{m_n}^*}{\\mathbb Z}_n(h)$ for $|h| \\le\\delta m_n^{1\/3}\/2 $\nfor all but a finite number of $n$ w.p. 1. By increasing the values of\n$C$ and $n_0$, if necessary, we may suppose that the right-hand side of\n(\\ref\n{eq:abc1}) (with $c=C$) is less than $\\varepsilon\/3$, that $P[|\\eta| > C]\n+ P[\\sup_{0 \\le t \\le1} m_n^{1\/6}|{\\mathbb E}_{m_n}(t)-{\\mathbb\nB}_{m_n}^0(t)| > C] \\le\\varepsilon\/3$, and that $L_{I_{m_n}}{\\mathbb Z}_n\n= L_{I_{m_n}^*}{\\mathbb Z}_n$ on $[-{1\\over2}\\delta m_n^{1\/3},\n{1\\over2}\\delta m_n^{1\/3}]$ with probability at least $1 -\n\\varepsilon\/3$ for all $n \\ge n_0$. We can also assume that $\\alpha:=\n8C^3\/\\gamma> 1$. Then, using Lemma \\ref{lem:abc} with $a = \\alpha\nm_n^{-{1\/6}}$ and $b = a^{-3}$, the following relations hold\nsimultaneously with probability at least $1-\\varepsilon$ for $n \\ge n_0$:\n\\begin{eqnarray*}\n|{\\mathbb B}_{m_n}[F_n(t_0) + s] - {\\mathbb B}_{m_n}[F_n(t_0)]|\n& \\le &\nC\\bigl(\\alpha m_n^{-{1\/6}} + \\alpha^{-3}\\sqrt{m_n}s^2\\bigr)\\qquad \\mbox{for\nall } s, \\\\\nL_{I_{m_n}}{\\mathbb Z}_n & = & L_{I_{m_n}^*}{\\mathbb Z}_n \\qquad\\mbox{on }\n\\biggl[-\\frac{\\delta}{2} m_n^{1\/3},\\frac{\\delta}{2} m_n^{1\/3}\\biggr],\n|\\eta| \\le C,\n\\end{eqnarray*}\nand\n\\[\n\\sup_{0 \\le t \\le1} m_n^{1\/6}|\n{\\mathbb E}_{m_n}(t) - {\\mathbb B}_{m_n}^0(t)| \\le C.\n\\]\nLet $B_n$ be the event that these four conditions hold. Then $P(B_n)\n\\ge1-\\varepsilon$ for $n \\ge n_0$, and from (\\ref{eq:boots_proc1}),\n$B_n$ implies\n\\begin{eqnarray}\\label{eq:boundZn1Crude}\n|{\\mathbb Z}_{n,1}(h)| & \\le & C \\{\\alpha+ \\alpha^{-3}m_n^{2\/3}\n[F_n(t_0+m_n^{-{1\/3}}h) - F_n(t_0)]^2 \\} + 2C \\nonumber\\\\\n&&{} + Cm_n^{1\/6}|F_n(t_0 + m_n^{-{1\/3}}h) -\nF_n(t_0)| \\\\\n&\\le& 4 C \\{\\alpha+ \\alpha^{-1}m_n^{2\/3}[F_n(t_0+m_n^{-{1\/3}}h) -\nF_n(t_0)]^2 \\}\\nonumber\n\\end{eqnarray}\nusing the inequalities $|F_n(t_0 + m_n^{-{1\/3}} h) - F_n(t_0)| \\le\n\\alpha m_n^{-{1\/6}} + \\alpha^{-1}m_n^{1\/6}[F_n(t_0 +\nm_n^{-{1\/3}} h) - F_n(t_0)]^2$ and $\\alpha> 1$. For sufficiently\nlarge $n$, using (\\ref{eq:cndtn4}), we have\n\\begin{eqnarray}\\label{eq:boundZn1}\n|\\mathbb{Z}_{n,1} (h)| & \\le& 4 C [\\alpha+ \\alpha^{-1}C^2\nm_n^{2\/3} (m_n^{-{1\/3}}|h| + m_n^{-{1\/3}})^2]\n\\nonumber\\\\\n& \\le& 4 C [\\alpha+ 2\\alpha^{-1} C^2 (h^2 + 1) ] \\\\\n& = & \\gamma h^2 + \\mathcal{C}\\nonumber\n\\end{eqnarray}\nfor $|h| \\le\\delta m_n^{1\/3}$ with $\\mathcal{C} = 4C\\alpha+\n8C^3\\alpha^{-1}$. Also, we can show that $|{\\mathbb Z}_{n,2}(h) -\nf'(t_0)h^2 \/2| \\le\\gamma h^2 + \\mathcal{C}$\nfor all $|h| \\le\\delta m_n^{1\/3}$ by (\\ref{eq:cndtn3}). Let $b_2\n> a_2$ be such that $- 5 \\gamma(a_2 + b_2)^2 + 6 \\gamma(a_2^{2} +\nb_2^2) - 8\\mathcal{C} > 0$.\n\nRecalling that $\\gamma= -f'(t_0)\/16$, $B_n$ implies\n\\[\n-10 \\gamma h^2 - 2\\mathcal{C} \\le{\\mathbb Z}_n(h) = {\\mathbb\nZ}_{n,1}(h) + {\\mathbb Z}_{n,2}(h) \\le- 6 \\gamma h^2 + 2\\mathcal{C}\n\\]\nfor $|h| \\le\\delta m_n^{1\/3}$ and sufficiently large $n$. Since\nthe right-hand side is concave, $B_n$ also implies\n$L_{I_{m_n}^*}{\\mathbb\nZ}_n(h) \\le- 6\\gamma h^2 + 2\\mathcal{C}$ for $|h| \\le\\delta m_n^{1\/3}$. Therefore, for\nsufficiently large $n$, using the upper bound on $L_{I_{m_n}^*}{\\mathbb\nZ}_n$, the lower bound on $\\mathbb{Z}_n$ obtained above, and\n$L_{I_{m_n}}{\\mathbb Z}_n(h) = L_{I_{m_n}^*}{\\mathbb Z}_n(h)$ for $|h|\n\\le\\delta m_n^{1\/3}\/2$ on $B_n$, and $[a_2,b_2] \\subset I_{m_n}^*$, we have\n\\begin{eqnarray*}\n&&2{\\mathbb Z}_n \\biggl( {a_2 + b_2 \\over2} \\biggr) -\n[L_{I_{m_n}}{\\mathbb Z}_n(a_2) + L_{I_{m_n}}{\\mathbb\nZ}_n(b_2) ] \\\\\n&&\\qquad\\ge- 5 \\gamma(a_2 + b_2)^2 + 6 \\gamma(a_2^{2} + b_2^2) - 8\\mathcal\n{C} > 0\n\\end{eqnarray*}\nwith probability at least $1-\\varepsilon$. Thus, $B_n$ implies $2\n{\\mathbb\nZ}_n[{1\\over2}(a_2 + b_2)] > L_{I_{m_n}}{\\mathbb Z}_n(a_2) +\nL_{I_{m_n}}{\\mathbb Z}_n(b_2)$ with probability at least $1-\\varepsilon$.\nSimilarly, $B_n$ implies that there is a $b_1 < a_1$ for which $2\n{\\mathbb Z}_n[{1\\over2}(a_1 + b_1)] > L_{I_{m_n}}{\\mathbb Z}_n(a_1) +\nL_{I_{m_n}}{\\mathbb Z}_n(b_1)$ with probability at least $1-\\varepsilon$.\nRelation (\\ref{eq:loc}) then follows from Lemma \\ref{lem:ww}. It is\nworth noting as a \\textit{remark} that $b_1, b_2$ do not depend on the\nsequence $F_n$.\n\nNext, consider (\\ref{eq:loc2}). Given a compact $J = [-b,b]$, let\n$c_{0}(\\omega)$ be the smallest positive integer such that for any $c\n\\ge c_{0}$, $L_{\\mathbb{R}} \\mathbb{Z}(h) = L_{[-c,c]} \\mathbb{Z} (h) $\nfor $h \\in J$. That $c_{0}$ exists and is finite w.p. 1 follows from\nLemma \\ref{lemma:slope_lcm}. Defining $W_c := L_{[-c,c]} \\mathbb{Z}$\nand $Y = L_{\\mathbb{R}} \\mathbb{Z}$, the event $\\{W_c \\ne Y \\mbox{ on }\nJ\\} \\subset\\{c_{o} > c\\}$. Now given any $\\varepsilon> 0$, there exist\n$c$ such that $P[c_{o} \\le c] > 1 - \\varepsilon$. Therefore,\n\\[\nP\\bigl[L_{\\mathbb{R}} \\mathbb{Z} = L_{[-c,c]} \\mathbb{Z} \\mbox{ on } J \\bigr] \\ge\nP[c_{o} \\le c] > 1 - \\varepsilon.\n\\]\n\\upqed\\end{pf*}\n\\begin{pf*}{Proof of Proposition \\ref{prop:edflcm}} First, consider\n${\\mathbb F}_n$. Let $0 < \\gamma< |f'(t_0)|\/2$ be given. There is a $0\n< \\delta< {1\\over2}t_0$ such that\n\\begin{eqnarray}\n\\label{eq:2diffF}\n\\bigl| F(t_0 + h) - F(t_0) - f(t_0)h - \\tfrac{1}{2} f'(t_0) h^2 \\bigr|\n\\le\\tfrac{1}{2} \\gamma h^2\n\\end{eqnarray}\nfor $|h| \\le2\\delta$. From the proof of Proposition \\ref{prop:loc},\nusing arguments similar to deriving (\\ref{eq:boundZn1Crude}) and (\\ref\n{eq:boundZn1}), we can show that\n\\[\n|({\\mathbb F}_n-F)(t_0+h)-({\\mathbb F}_n-F)(t_0)| < \\tfrac{1}{2}\\gamma\nh^2 + C n^{-{2 \/3}}\n\\]\nfor $|h| \\le2 \\delta$ with probability at least $1 - \\varepsilon$ for\nsufficiently large $n$. Therefore, by adding and subtracting $F(t_0+h)\n- F(t_0)$ and using (\\ref{eq:2diffF}),\n\\begin{equation}\n\\label{eq:edf}\n\\bigl| {\\mathbb F}_n(t_0 + h) - {\\mathbb F}_n(t_0) - f(t_0)h - \\tfrac\n{1}{2} f'(t_0) h^2 \\bigr|\n\\le\\gamma h^2 + C n^{-{2\/3}}\n\\end{equation}\nfor $|h| \\le2 \\delta$ with probability at least $1-\\varepsilon$ for\nlarge $n$.\n\nNext, consider $\\tilde{F}_n$. Let $B_n$ denote the event that (\\ref\n{eq:edf}) holds. Then $P(B_n)$ is eventually\nlarger than $1 - \\varepsilon$ and on $B_n$, we have\n\\[\n{\\mathbb F}_n(t_0+h) - \\mathbb{F}_n(t_0) - f(t_0) h \\le\\bigl\\{ \\gamma\n- \\tfrac{1}{2} |f'(t_0)| \\bigr\\} h^2 + Cn^{-{2\/3}}\n\\]\nfor $|h| \\le2 \\delta$. Let $E_n$ be the event that $\\tilde{F}_n(h) =\nL_{[t_0-2\\delta,t_0+2\\delta]}{\\mathbb F}_n(h)$ for $h \\in[t_0-\n\\delta,t_0+ \\delta]$. Then by Lemma \\ref {lem:loc}, $P(E_n) \\ge1 -\n\\varepsilon$, for all sufficiently large $n$. Taking concave majorants\non either side of the above display for $|h| \\le2 \\delta$ and noting\nthat the right-hand side of the display is already concave, we have:\n${\\tilde F}_n(t_0+h) - \\mathbb{F}_n(t_0) - f(t_0) h \\le\\{ \\gamma-\n\\frac{1}{2} |f'(t_0)| \\} h^2 + C n^{-{2\/3}}$, for $|h| \\le\\delta$ on\n$B_n \\cap E_n$. Setting $h = 0$ shows that on $E_n \\cap B_n$, $\\tilde\nF_n(t_0) - \\mathbb{F}_n(t_0) \\le C n^{-{2\/3}}$. Now, as ${\\mathbb\nF}_n(t_0) \\le\\tilde{F}_n(t_0)$, it is also the case that on $E_n \\cap\nB_n$, for $|h| \\le\\delta$,\n\\begin{equation}\n\\label{eq:cncvbnd2}\n{\\tilde F}_n(t_0+h) - \\tilde F_n(t_0) - f(t_0) h \\le\\bigl\\{ \\gamma-\n\\tfrac{1}{2} |f'(t_0)| \\bigr\\} h^2 + C n^{-{2\/3}}.\n\\end{equation}\nFurthermore on $E_n \\cap B_n$,\n\\begin{eqnarray}\n\\label{eq:cncvbnd3}\n&& \\tilde F_n(t_0 + h) - \\tilde F_n(t_0) - f(t_0) h - \\tfrac{1}{2}\nf'(t_0) h^2 \\nonumber\\\\\n&&\\qquad \\ge \\mathbb F_n(t_0 + h) - \\{\\mathbb F_n(t_0) + C n^{-{2\/3}}\\}\n- f(t_0) h - \\tfrac{1}{2} f'(t_0) h^2\n\\\\\n&&\\qquad \\ge -\\gamma h^2 - 2 C n^{-{2\/3}}.\\nonumber\n\\end{eqnarray}\nTherefore, combining (\\ref{eq:cncvbnd2}) and (\\ref{eq:cncvbnd3}),\n\\[\n\\bigl| {\\tilde F}_n(t_0 + h) - {\\tilde F}_n(t_0) - f(t_0)h - \\tfrac\n{1}{2} f'(t_0) h^2 \\bigr| \\le\\gamma h^2 + 2C n^{-{2\/3}}\n\\]\nfor $|h| \\le\\delta$ with probability at least $1 - 2\\varepsilon$ for\nlarge $n$.\n\\end{pf*}\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe main purpose of this work is to expose the role of gravity in the dynamics of particles in a rotating trap. In addition, we present a complete analysis of the stability regions for a rotating trap in 3D. We prove that in the generic case there are three separate regions of stability with different characteristics. Gravity induced resonances are relevant only if they occur in the regions of stability, otherwise, they are swamped by the exponential behavior of trajectories.\n\nHarmonic traps are often used in optics and atomic physics (especially in the form of TOP traps \\cite{petrich} in the study of Bose-Einstein condensates) and yet a complete theory of these devices has not been developed. The solution to the problem of a trap rotating around one of the trap axes is effectively two-dimensional and its solution has been known for at least hundred years. In the classic textbook on analytical dynamics by Whittaker \\cite{whitt} we find a solution of a mathematically equivalent problem of small oscillations of ``a heavy particle about its position of equilibrium at the lowest point of a surface which is rotating with constant angular velocity about a vertical axis through the point''. The quantum-mechanical counterpart of the Whittaker problem has also been completely solved \\cite{linn,oktel} and the statistical mechanics of a classical gas was studied in \\cite{gd}.\n\nIn this work we present a complete solution to the problem of the motion of a particle moving in a most general anisotropic rotating harmonic trap in 3D and in the presence of gravity. This is an exactly soluble problem but technical difficulties apparently served so far as a deterrent in developing a full description. A full description of the particle dynamics in a rotating anisotropic trap in three dimensions so far has not been given despite a new significance of this problem brought about by experimental and theoretical studies of Bose-Einstein condensates and the accompanying thermal clouds in rotating traps \\cite{rzs,mad,sc,gg,ros,cozz,abo,gd}. Explicit formulas describing the complete mode structure in the three-dimensional case are indeed quite cumbersome \\cite{cmm} because we deal here with third-order polynomials and on top of that they have rather complicated coefficients. However, many important features may be exhibited without straining the reader's patience. In particular, we can identify various stability regions for an arbitrary orientation of the angular velocity and we can give conditions for a resonance.\n\nThe standard arrangement \\cite{rzs,mad,sc,gg,ros,cozz,abo,gd} is to choose a vertical axis of rotation of the trap. Slight tiltings of this axis were introduced to excite the scissors modes \\cite{gds,marago,smith}. However, for such very small tilting angles the effects described in the present paper would not be noticeable. In the case of a vertical axis of rotation, there are no resonances. The only effect of gravity is a displacement of the equilibrium position. The situation completely changes and new phenomena will occur when the axis of rotation is tilted away from the vertical position. In this generic case, for every anisotropic three-dimensional trap there exist two (not three as one might expect) characteristic frequencies at which resonances occur. The motion in a trap that is rotating at a {\\em resonant} frequency will become unbounded and all particles will be expelled from the trap. The position of the resonance does not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity.\n\nAll our results are valid not only for a single particle but also for the center of mass motion in many-body (classical or quantum) theory since for all quadratic Hamiltonians the center of mass motion completely separates from the internal motion \\cite{kohn,dob,cmm}. Therefore, a trap rotating at the resonant frequency will not hold the Bose-Einstein condensate. Owing to the linearity of the equations of motion for a harmonic trap, all conclusions hold both in classical and in quantum theory. A resonant behavior caused by an application of a static force may seem counterintuitive, but it is explained by the fact that in a rotating frame the force of gravity acts as a {\\em periodically changing} external force.\n\n\\section{Equations of motion}\n\nThe best way to analyze the behavior of particles in a uniformly rotating trap is to first perform the transformation to the rotating frame. In this frame the harmonic trap potential is frozen but the force of gravity is rotating with the angular velocity of the trap rotation. In the rotating frame the Hamiltonian has the form\n\\begin{equation}\\label{ham}  {\\cal H} = \\frac{\\bm{p}^{2}}{2m}  + \\bm{r}\\!\\cdot\\!\\hat{\\Omega}\\!\\cdot\\!\\bm{p}  + \\frac{m}{2}\\bm{r}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{r} - m\\bm{r}\\!\\cdot\\!\\bm{g}(t).\n\\end{equation}\nThe potential matrix $\\hat{V}$ is symmetric and positive definite. The eigenvalues of this matrix are the squared frequencies of the oscillations in the non-rotating trap. The angular velocity matrix $\\hat{\\Omega}$ is related to the components of the angular velocity vector through the formula ${\\Omega}_{ik}=\\epsilon_{ijk}\\Omega_j$. The vector of the gravitational acceleration $\\bm{g}(t)$, as seen in the rotating frame, can be expressed in the form\n\\begin{eqnarray}\\label{rotg}\n{\\bm g}(t) = {\\bm g}_{\\parallel} + {\\bm g}_{\\perp}\\cos(\\Omega t) - ({\\bm n}\\times{\\bm g}_{\\perp})\\sin(\\Omega t),\n\\end{eqnarray}\nwhere ${\\bm n}$ denotes the direction and $\\Omega$ denotes the length of the angular velocity vector ${\\bm\\Omega}$. The parallel and the transverse components of the gravitational acceleration vector ${\\bm g}={\\bm g}(0)$ are defined as ${\\bm g}_{\\parallel} = {\\bm n}({\\bm n}\\!\\cdot\\!{\\bm g})$ and ${\\bm g}_{\\perp} = {\\bm g}-{\\bm n}({\\bm n}\\!\\cdot\\!{\\bm g})$, respectively. Note that the time-dependent part vanishes when the rotation axis is vertical.\n\nThe equations of motion determined by the Hamiltonian (\\ref{ham}) have the following form\n\\begin{subequations}\n\\begin{eqnarray} \\label{3dimeq}\n\\frac{d{\\bm r}(t)}{dt} &=& \\frac{{\\bm p}(t)}{m} - \\hat{\\Omega}\\!\\cdot\\!{{\\bm r}(t)},\\\\\n\\frac{d{\\bm p}(t)}{dt} &=& -m\\hat{V}\\!\\cdot\\!{{\\bm r}(t)} - \\hat{\\Omega}\\!\\cdot\\!{{\\bm p}(t)} + m {\\bm g}(t).\n\\end{eqnarray}\n\\end{subequations}\nThese equations describe an oscillator in a rotating frame displaced by a constant force (the longitudinal part of ${\\bm g}$) and driven by a periodic force (the transverse part of ${\\bm g}$). It is convenient to rewrite the expression (\\ref{rotg}) as a real part of a complex function\n\\begin{eqnarray}\\label{rotg1}\n {\\bm g}(t) = \\Re\\left({\\bm g}_{\\parallel}+({\\bm g}_{\\perp} +  i({\\bm n}\\times{\\bm g}_{\\perp}))e^{i\\Omega t}\\right).\n\\end{eqnarray}\nIn compact notation Eqs.~(\\ref{3dimeq}) have the form\n\\begin{equation} \\label{eqnmot}\n\\frac{d{\\cal R}(t)}{dt} = \\hat{\\cal M}(\\Omega)\\!\\cdot\\!{\\cal R}(t) + \\Re({\\cal G}_{\\parallel} + {\\cal G}_{\\perp}{e}^{i\\Omega t}),\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{\\cal R}(t) &=& \\left(\\begin{array}{c} \\bm{r}(t) \\\\ \\bm{p}(t) \n\\end{array}\\right),\\;\\;\n\\hat{\\cal M}(\\Omega) = \\left(\\begin{array}{cc} - \\hat{\\Omega} & m^{-1}\\hat{I} \\\\ -m\\hat{V} & - \\hat{\\Omega}\\end{array}\\right), \\\\\n{\\cal G}_{\\parallel} &=&  m\\left(\\begin{array}{c} 0 \\\\ \\bm{g}_{\\parallel} \\end{array}\\right),\\;\\;\n{\\cal G}_{\\perp} = m\\left(\\begin{array}{c} 0 \\\\ \\bm{g}_{\\perp} + i(\\bm{n}\\times\\bm{g}_{\\perp}) \\end{array}\\right). \n\\end{eqnarray}\nWe shall now replace the equations of motion by their complex counterpart\n\\begin{equation} \\label{eqnmotc}\n\\frac{d{\\cal W}(t)}{dt} = \\hat{\\cal M}(\\Omega)\\!\\cdot\\!{\\cal W}(t) + {\\cal G}_{\\parallel} + {\\cal G}_{\\perp}{e}^{i\\Omega t}.\n\\end{equation}\nThe physical trajectory in phase space is described by the real part of the complex vector ${\\cal W}(t)$. Let us introduce a basis of six eigenvectors of $\\hat{\\cal M}(\\Omega)$\n\\begin{equation}\n\\hat{\\cal M}(\\Omega){\\cal X}_{k} \n= i \\omega_{k}(\\Omega){\\cal X}_{k},\\qquad k = 1,\\ldots, 6\n\\end{equation}\nand expand ${\\cal W}(t)$ and ${\\cal G}(t)$ in this base as follows\n\\begin{eqnarray}\\label{expan}\n{\\cal W}(t) &=&\\sum_{k=1}^{6} \\alpha^k(t) \\, {e}^{i\\omega_k(\\Omega)t} \\, {\\cal X}_k, \\\\\n{\\cal G}_\\parallel &=& \\sum_{k=1}^{6} \\gamma^k_{\\parallel}\\, {\\cal X}_{k},\\;\\;\\;\n{\\cal G}_\\perp = \\sum_{k=1}^{6} \\gamma^k_{\\perp}\\, {\\cal X}_{k}. \n\\end{eqnarray}\nOwing to a simple block structure of $\\hat{\\cal M}(\\Omega)$, the basis vectors ${\\cal X}_{k}$ can be determined by reducing effectively the problem to three dimensions. We use this method in the Appendix B to determine the resonant solution. \n\nThe equation of motion (\\ref{eqnmotc}) can be rewritten now as a set of equations for the coefficient functions $\\alpha^k(t)$\n\\begin{equation} \\label{alphaeq}\n\\frac{d{\\alpha}^k(t)}{dt} = \\gamma^{k}_{\\parallel}{e}^{-i \\omega_k(\\Omega)t} + \\gamma^{k}_{\\perp}{e}^{i(\\Omega-\\omega_k(\\Omega))t},\\;\\; k = 1,\\ldots,6.\n\\end{equation}\nIt is clear now that the mode amplitude $\\alpha_k(t)$ will grow linearly in time --- the signature of a resonance --- whenever either one of the two terms on the right hand side becomes time independent. This happens to the first term if one of the frequencies $\\omega_k(\\Omega)$ vanishes but the corresponding coefficient $\\gamma^{k}_{\\parallel}$ does not vanish. This case is not interesting, since it means that we are just at the border of the lower instability region and the trap is not holding particles, as discussed in the next section. The second term becomes time independent when the angular velocity of trap rotation $\\Omega$ satisfies the resonance condition $\\Omega=\\omega_k(\\Omega)$ and, of course, $\\gamma^{k}_{\\perp} \\neq 0$. This resonance {\\em is different} from a resonance in a standard periodically driven oscillator. In the present case the characteristic frequencies of the trap depend on the frequency $\\Omega$ of the driving force. Therefore, the position of the resonance has to be determined selfconsistently. A full description of these gravity induced resonances requires the knowledge of the behavior of $\\omega_k(\\Omega)$'s as functions of $\\Omega$. In particular, it is important to know whether a resonance occurs in a region where the system undergoes stable oscillations. This will be discussed in the next Section.\n\n\\section{Regions of stability}\n\nThe stability of motion for a harmonic oscillator is determined by the values of its characteristic frequencies $\\omega$ --- the roots of the characteristic polynomial. In the present case, these frequencies are determined by the characteristic equation for the matrix $\\hat{\\cal M}(\\Omega)$\n\\begin{equation}\n\\mathrm{Det}\\left\\{ \\hat{\\cal M}(\\Omega) - i\\omega \\right\\} = 0.\n\\end{equation}\nThe characteristic polynomial is tri-quadratic\n\\begin{equation} \\label{charpoly}\nQ(\\chi)=\\chi^3 + A\\,\\chi^2 + B\\,\\chi + C,\\;\\;\\;\\chi=\\omega^2,\n\\end{equation}\nwhere the coefficients $A$, $B$ i $C$ can be expressed in a rotationally invariant form \\cite{cmm}\n\\begin{eqnarray}\nA &\\!=\\!&-2\\Omega^2 - \\mathrm{Tr}\\{\\hat{V}\\},\\nonumber\\\\\nB &\\!=\\!&\\Omega^4\\!+\\!\\Omega^2(3\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\mathrm{Tr}\\{\\hat{V}\\})\\!\n+\\!\\frac{\\mathrm{Tr}\\{\\hat{V}\\}^2\\!-\\!\\mathrm{Tr}\\{\\hat{V}^2\\}}{2},\\nonumber\\\\\nC &\\!=\\!&{\\Omega}^2(\\mathrm{Tr}\\{\\hat{V}\\}\\!\n-\\!{\\Omega}^2)\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\Omega^2\\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\mathrm{Det}\\{\\hat{V}\\}.\\qquad\n\\end{eqnarray}\nStable oscillations take place when all characteristic $\\omega$'s are real. This means that all three roots of the polynomial $Q(\\chi)$ must be real and positive. Without rotation, when $\\Omega=0$, the three roots of $Q(\\chi)$ are equal to the eigenvalues of the potential matrix $\\hat{V}$. We have then a simple system of three harmonic oscillators vibrating independently along the principal directions of the trap. As $\\Omega$ increases, our system will, in general, go through two regions of instability: the lower region when one of the roots of $Q(\\chi)$ is negative and the upper region when two roots are complex. We shall exhibit this behavior by plotting the zero contour lines of $Q(\\chi)$ in the $\\Omega\\chi$-plane. We assume that the trap potential and the direction of rotation are fixed and we treat the characteristic polynomial $Q(\\chi)$ as a function of $\\Omega$ and $\\chi$ only. Contour lines representing the zeroes of $Q(\\chi)$ in the generic case are shown in Fig.~\\ref{fig1}. There is a region of $\\Omega$, where only one real root of $Q(\\chi)$ exists. However, this region is bounded, so for sufficiently large $\\Omega$ the system is always stable.\n\nIt has been argued in Ref.~\\cite{cmm} that there is always a region of instability when one of the roots of $Q(\\chi)$ is negative. The corresponding modes grow exponentially with time. As seen in Fig.~\\ref{fig1}, this region of instability is bounded by the two values $\\Omega_{1,2}$ at which the curve crosses the vertical axis. These values are given by the zeroes of $C$, treated as a biquadratic expressions in $\\Omega$\n\\begin{equation}\\label{zeroes}\n\\Omega_{1,2} = \\sqrt{\\frac{b\\pm\\sqrt{b^2-4ac}}{2a}},\n\\end{equation}\nwhere $a=\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n},\\;\nb=\\mathrm{Tr}\\{\\hat{V}\\}\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\n-\\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n}$, and $c=\\mathrm{Det}\\{\\hat{V}\\}$. Since $a, b$, and $c$ are positive and $b^2\\geq 4ac$, both values $\\Omega_{1}$ and $\\Omega_{2}$ are real. A degenerate case is possible, when $\\Omega_{1}=\\Omega_{2}$ then the region of instability shrinks to zero. In order to determine, when this can happen, we may use the (explicitly non-negative) representation of the discriminant $b^2-4ac$ given in Ref.~\\cite{cmm}. Assuming for definitness that $V_x<V_y<V_z$, we find that this happens in two cases: when the trap is not fully anisotropic ($V_x=V_y$ or $V_y=V_z$) or the axis of rotation lies in the $xz$-plane and its azimuthal angle satisfies the condition $\\sin^2\\theta=(1-V_x\/V_y)\/(1-V_x\/V_z)$. The second possibility has not been noticed in Ref.~\\cite{cmm}.\n\nGraphical representation of the solutions for $n_x=1$, $n_y=n_z=0$ is shown in Fig.~\\ref{fig2}. In this plot only one instability region is present, where one of the roots of (\\ref{charpoly}) is negative. Owing to the stabilizing effect of the Coriolis force, for fast rotations the system becomes again stable. We can also see that one of the characteristic frequencies remains constant, it does not vary with $\\Omega$. It is so, because the rotation does not influence the motion in the direction of the rotation axis. This is the degenerate case described by Whittaker and thoroughly studied in connection with the BEC traps \\cite{linn,oktel,gd,ros}. In this case there is only one instability region, where the square of the frequency is negative. When the direction of angular velocity is not parallel to one of the axes of the trap, a second kind of instability appears. In addition to the region of (purely exponential) instability, described before, when one root of $Q(\\chi)$ was negative there is also, in general, an additional region of (oscillatory) instability where two roots are complex. Since the coefficients of the characteristic polynomial are real, the square of the second frequency is complex conjugate to the first one. In this case the instability has the form of expanding oscillations. In Fig.~\\ref{fig3} we show how the second kind of instability develops when the rotation axis is tilted away a little bit from the $z$ direction.\n\nWe will show now, that the second kind of instability does not occur in the most common case, when the rotation vector is along one of the axes of the trap. In particular, it can never happen in a two dimensional trap. Without any loss of generality, we may assume that the direction of rotation is along the $z$ axis. In this degenerate case the characteristic polynomial (\\ref{charpoly}) factorizes because the motion in the $z$ direction is not influenced by rotation,\n\\begin{eqnarray}\n\\label{factor}\nQ(\\chi) = (\\chi - V_z)\\hspace{6cm}\\nonumber\\\\\n\\times(\\chi^2-\\chi(2\\Omega^2+V_x+V_y)+\\Omega^4\n-\\Omega^2(V_x+V_y)+V_x V_y).\\nonumber\\\\\n\\end{eqnarray}\nThe factor quadratic in $\\chi$ has real zeros if its discriminant $\\Delta$ is positive and this is, indeed, the case since\n\\begin{eqnarray} \\label{delta}\n\\Delta = 8\\Omega^2(V_x+V_y)+(V_x-V_y)^2 \\geq 0.\n\\end{eqnarray}\nIt follows, however, from the topology of the curves representing the zeros that the second kind of instability always exists if the direction of rotation does not coincide with one of the axes of the harmonic trap. \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig1.eps}\n\\caption{The contour lines in this figure represent the zeroes of the characteristic polynomial (\\ref{charpoly}) for $V_x=1$, $V_y=2$, $V_z=3$ and $n_x=1\/\\sqrt{3}$, $n_y=1\/\\sqrt{3}$, $n_z=1\/\\sqrt{3}$ plotted as functions of the magnitude of angular velocity $\\Omega$ and the square of the characteristic frequency $\\chi=\\omega^2$. In addition to the lower region of instability (between dashed lines) where one of the roots of (\\ref{charpoly}) is negative, there is also an upper region (between dotted lines) where there exists only one real root (determined by the continuation of the line that begins at $\\chi=3$, not seen in this figure because it corresponds to a very large value of $\\chi$). Since there is no absolute scale of frequencies involved in the analysis of stability regions, we have chosen the lowest trap frequency as a unit.}\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig2.eps}\n\\caption{This plot is for the same trap as in Fig.~\\ref{fig1} but for the rotation around the trap axis $n_x=0$, $n_y=0$, $n_z=1$. In this degenerate case there exists only one region of instability, when one of the roots of (\\ref{charpoly}) is negative.} \\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig3.eps}\n\\caption{This plot shows how the degenerate case (Fig.~\\ref{fig2}) merges with the general case (Fig.~\\ref{fig1}). Here we have chosen the following values of the parameters: $V_x=1$, $V_y=2$, $V_z=3$ and $n_x=\\sin(1\/10)$, $n_y=0$, $n_z=\\cos(1\/10)$.} \\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig4.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{2\\pi}{5})$, $n_y=0$, $n_z=\\cos(\\frac{2\\pi}{5})$. Two horizontal lines enclose the lower region of instability. Both resonant frequencies lie in the lower region of stability.} \\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig5.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{\\pi}{4})$, $n_y=0$, $n_z=\\cos(\\frac{\\pi}{4})$. Higher resonant frequency lies in the lower region of instability.} \\label{fig5}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig6.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{\\pi}{60})$, $n_y=0$, $n_z=\\cos(\\frac{\\pi}{60})$. Resonant frequencies lie in two different regions of stability.} \\label{fig6}\n\\end{figure}\n\n\\section{Gravity induced resonances}\n\nThere are, in general, two resonant values of $\\Omega$. They occur when $\\chi=\\Omega^2$. In this case, from Eq.~(\\ref{charpoly}) we get a biquadratic equation for the resonant values of $\\Omega$:\n\\begin{equation} \\label{reseq}\n D \\Omega^4 + E \\Omega^2 +F = 0,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nD &=& -2(\\,\\mathrm{Tr}\\{\\hat{V}\\}-\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\,), \\nonumber \\\\\nE &=& \\frac{\\mathrm{Tr}\\{\\hat{V}\\}^2\\!-\\!\\mathrm{Tr}\\{\\hat{V}^2\\}}{2}\n+\\mathrm{Tr}\\{\\hat{V}\\}\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\! - \\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n},\\nonumber \\\\\nF &=& -\\mathrm{Det}\\{\\hat{V}\\}.\n\\end{eqnarray}\nThe properties of the two solutions $\\Omega_{\\pm}^2$ of Eq.~(\\ref{reseq}) are described in the Appendix A. We may also analyze the resonance condition graphically by superposing the parabola $\\chi=\\Omega^2$ (shown as a thick line in Figs.~\\ref{fig4}--\\ref{fig7}) on the plots of characteristic frequencies. The lower value is more interesting because, as shown in the Appendix A, it always falls into the range of the first region of stable oscillations where we would expect the confinement of particles in the trap. The upper value may fall in the lower region of stability (Fig.~\\ref{fig4}), in the lower region of instability (Fig.~\\ref{fig5}), or in the higher region of stability (Fig.~\\ref{fig6}). A degenerate case is also possible (cf. Fig.~\\ref{fig7}) when the two resonance frequencies coincide but this happens only under very special circumstances (see Appendix A). The existence of only two resonant frequencies (we would expect three resonant frequencies for a three dimensional oscillator) clearly shows that we are dealing here with a more complex dynamical system than a simple driven harmonic oscillator.\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig7.eps}\n\\caption{In the degenerate case, when two resonant frequencies coincide, their common value is also equal to the lowest frequency of the trap. In this case the trap parameters are: $V_x = 1, V_y = 10\/3, V_z = 5$, and $n_x = 1$.}\\label{fig7}\n\\end{figure}\n\nThe essential difference between the resonant and the nonresonant behavior is illustrated in Figs.~\\ref{fig8}-\\ref{fig10}. The role of gravity is best seen by comparing Fig.~\\ref{fig8} (with gravity) and Fig.~\\ref{fig11} (gravity switched off). The calculations were performed in the coordinate system in which the potential matrix $\\hat V$ is diagonal and it was assumed that the force of gravity at $t=0$ is directed along the $z$ axis of the trap. All four figures were generated for the same trap and under the same initial conditions: the particle is initially placed in the center of the trap and it is given the initial velocity of 1 cm\/s in the $x$ direction. These simple initial conditions result in the excitation of all the modes of the oscillator. However, at resonance the mode growing linearly with $t$ dominates the time evolution. This can be seen by comparing, for the same parameters of the trap and at the resonant frequency $\\Omega\/2\\pi = 6.49421$ Hz, the motion depicted in Fig.~\\ref{fig8}, that was calculated numerically, with the motion depicted in Fig.~\\ref{fig12}. The second plot was obtained from an analytic solution of the equations of motion (\\ref{eqnmotc}), that contains only the resonant part. This analytic solution is derived in the Appendix B. Both plots are essentially the same except for very small differences that are due to the fact that the initial conditions assumed in Fig.~\\ref{fig8} require some admixture of the nonresonant solutions while the analytic solution does not contain any nonresonant pieces.\n\nGravity induced resonances are significant only if they occur in the region of stability. In all regions of instability, where the trajectories exhibit exponential growth, the resonances cannot be detected.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig8.eps}\n\\caption{The trajectory of a particle in the coordinate frame rotating with the trap. The characteristic frequencies of the trap in the $x$, $y$, and $z$ directions are 10 Hz, 15 Hz, and 20 Hz, respectively. The angular velocity vector has the direction $(1,1,1)$ and its length $\\Omega\/2\\pi = 6.49421$ Hz satisfies the resonance condition, as calculated in the Appendix A. The distances in this plot are measured in centimeters. The direction of the gravitational force ($g=9.81 {\\rm m s}^{-2}$) is assumed to coincide at $t=0$ with the $z$-axis of the trap. We can see that the amplitude of oscillations increases linearly with time.}\\label{fig8}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig9.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the value of the angular velocity  $\\Omega\/2\\pi = 4$ Hz is below the resonance. The scale in this plot is reduced by a factor of 10 as compared to Fig.~\\ref{fig8}.}\\label{fig9}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig10.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the value of the angular velocity $\\Omega\/2\\pi = 9$ Hz is above the resonance. The scale in this plot is reduced by a factor of 10 as compared to Fig.~\\ref{fig8}.}\\label{fig10}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig11.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the gravitational field has been turned off. The angular velocity $\\Omega\/2\\pi = 6.49421$ Hz has the resonant value. The scale in this plot is reduced by a factor of 100 as compared to Fig.~\\ref{fig8}. Clearly, there is no sign of any resonant behavior.}\\label{fig11}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig12.eps}\n\\caption{The trajectory of a particle described by the analytic formulas given in the Appendix B. The trap parameters and the angular velocity are the same as in Fig.~\\ref{fig8}.}\\label{fig12}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe have analyzed the stability of motion in an anisotropic, rotating harmonic trap in 3D. We have found that, in general, there are three regions of stability. The second region and the third region merge only when the rotation axis coincides with the one of the trap axes. We have demonstrated the presence of resonances in a rotating harmonic trap subjected to the force of gravity. The nature of these resonances is different from the standard forced harmonic oscillator since the resonant frequencies must be determined selfconsistently. The resonances occur at two rotation rates but they exist only when the rotation axis is tilted away from the vertical direction. However, the rotation axis can still be directed along one of the trap axes. The lower resonance always falls in the region of stable oscillations where as the higher resonance may fall in the lower region of stability, in the first region of instability, or in the higher region of stability. Resonant rotation rates depend solely on the properties of the trap and not on the masses of particles. The resonances cause the escape of a particle, or the center of mass for a collection of interacting particles, from the trap. In view of recent experiments in which the axis of rotation was tilted away from the trap axis \\cite{marago,smith}, it should be possible to confirm experimentally the existence of the gravity-induced resonances for Bose-Einstein condensates in rotating traps. In particular, the presence of resonances makes it possible to quickly expel all particles from the trap.\n\nThis research was supported by a grant from the Polish Ministry of Scientific Research and Information Technology under the Grant No. PBZ-MIN-008\/P03\/2003.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}