Berry-Esseen bounds for the chi-square distance in the Central Limit Theorem: a Markovian approach

TL;DR

This paper introduces a new Markovian approach to establish Berry-Esseen bounds for the convergence rate of sums of independent random variables to the normal distribution in chi-square distance, relaxing the identical distribution assumption.

Contribution

It provides a novel proof leveraging spectral analysis of non-reversible Markov operators, achieving optimal convergence rates under weaker assumptions.

Findings

Established optimal convergence rate in chi-square distance

Extended results to non-identically distributed variables

Utilized spectral properties of Markov transition operators

Abstract

This article presents a new proof of the rate of convergence to the normal distribution of sums of independent, identically distributed random variables in chi-square distance, which was also recently studied in \cite{BobkovRenyi}. Our method consists of taking advantage of the underlying time non-homogeneous Markovian structure and studying the spectral properties of the non-reversible transition operator, which allows to find the optimal rate in the convergence above under matching moments assumptions. Our main assumption is that the random variables involved in the sum are independent and have polynomial density, interestingly, our approach allows to relax the identical distribution hypothesis.