Strong central limit theorem for isotropic random walks in \({\mathbb{R}^d}\)
type: Item
instance of: scholarly article
title: Strong central limit theorem for isotropic random walks in \({\mathbb{R}^d}\)
zbMATH Open document ID: 1236.60045
author: Piotr Graczyk
author: Jean-Jacques Loeb
author: Tomasz Żak
publication date: 2011-11-07T00:00:00Z
review text: Let \(d\geq2\), and let \(\sigma_d\) and \(\mu_d\) be the uniform distributions on the unit sphere and unit ball in \(\mathbb R^d\) respectively. Consider the associated normalized sums \(\tilde S_n:=n^{-1/2}\cdot \sum_{k=1}^n X_k\) for iid random variables \(X_k\) on \(\mathbb R^d\) with the distributions \(\sigma_d\) or \(\mu_d\) . The authors prove that the associated Lebesgue densities have a strong upper bound of the form \(c_d\cdot e^{-d|x|^2/2}\) independent of \(n\) for some constant \(c_d\) for \(\sigma_d\); for \(\mu_d\) and powers of \(\sigma_d\) and \(\mu_d\), a corresponding upper bound is constructed. These upper bounds lead to a strong version of a CLT for \(E(f(\tilde S_n))\) for \(n\to\infty\) for functions \(f\) which are integrable w.r.t. \(c_d\cdot e^{-d|x|^2/2}dx\). Moreover, the bounds have applications to strong hypercontractivity inequalities and strong logarithmic Sobolev inequalities for these isotropic random walks.
reviewed by: Michael Voit
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 5968711
zbMATH Keywords: isotropic random walks
zbMATH Keywords: central limit theorems
zbMATH Keywords: Gaussian upper bounds
zbMATH Keywords: logarithmic Sobolev inequality
MaRDI profile type: Publication
full work available at URL: s00440-010-0295-6
OpenAlex ID: W2073478876
cites work: On logarithmic Sobolev inequalities. With a preface of Dominique Bakry and Michel Ledoux
cites work: Q5558293
cites work: Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn.
cites work: Q4097679
cites work: On Gaussian and Bernoulli covariance representations
cites work: Dimension free and infinite variance tail estimates on Poisson space
cites work: Uniform Central Limit Theorems
cites work: Q3138785
cites work: Q3959189
cites work: Hypercontractivity for log-subharmonic functions
cites work: Q3084090
cites work: Logarithmic Sobolev Inequalities
cites work: Q4284289
cites work: Gaussian estimates for Markov chains and random walks on groups
cites work: Remarks on deviation inequalities for functions of infinitely divisible random vectors
cites work: Q4445168
cites work: Q4327751
cites work: An approximation of partial sums of independent RV'-s, and the sample DF. I
cites work: Extremal probabilistic problems and Hotelling's \(T^ 2\) test under a symmetry condition
cites work: Exact inequalities for sums of asymmetric random variables, with applications
cites work: Q5640160
cites work: The missing factor in Hoeffding's inequalities
DOI: 10.1007/S00440-010-0295-6
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 5968711