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Register a new userHigh-dimensional limit theorems for random vectors in $ell_p^n$-balls. II
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In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle when the underlying distribution of the random vectors belongs to a general class introduced by Barthe, Guedon, Mendelson, and Naor. It includes the normalized volume and the cone probability measure as well as projections of these measures as special cases. Two new applications to random and non-random projections of $ell_p^n$-balls to lower-dimensional subspaces are discussed as well. The text is a continuation of [Kabluchko, Prochno, Thale: High-dimensional limit theorems for random vectors in $ell_p^n$-balls, Commun. Contemp. Math. (2019)].
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Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. For fixed $p in (1,infty)$, let $(X^{(n,p)})$ and $(theta^n)$ be independent sequences of random vectors with $theta^n$ distributed according to the normalized cone measure on the unit $ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $ell_p^n$ sphere. For almost every sequence of projection directions $(theta^n)$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{n,p}$ onto $theta^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})$ is replaced with $(mathscr{X}^{(n,p)})$, where $mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(theta^n)$ that encodes geometric information. Moreover, although the (quenched) large deviation rate functions for the sequences of random projections of $(X^{(n,p)})$ and $(mathscr{X}^{(n,p)})$ are known to coincide, it is shown that the prefactor distinguishes between these two cases. The results on the one hand provide quantitative estimates of tail probabilities of random projections of $ell_p^n$ balls and spheres, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting.
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell of radius $sqrt{n}$ times the asymptotic value of $n^{-1/2}left(mathbb Eleft[| X_n|_2^2right]right)^{1/2}$ (as $ntoinfty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {bf 495}(1) 2021, 1--19].
Let $r=r(n)$ be a sequence of integers such that $rleq n$ and let $X_1,ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $rsim alpha n$ for some $0 < alpha < 1$).
We obtain explicit error bounds for the $d$-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a non-linear statistic of independent random variables or a sum of $n$ locally dependent random vectors. We assume the approximating normal distribution has a non-singular covariance matrix. The error bounds vanish even when the dimension $d$ is much larger than the sample size $n$. We prove our main results using the approach of Gotze (1991) in Steins method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of $n$ independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a $log n$ factor. We also discuss an application to multiple Wiener-It^{o} integrals.
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $log Z_n$.
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