A Berry-Esseen bound with (almost) sharp dependence conditions

TL;DR

This paper establishes near-optimal dependence conditions under which the normalized sum of a stationary sequence converges to a normal distribution at the optimal rate, applicable to various processes including dynamical systems.

Contribution

It provides (almost) sharp dependence conditions for the Berry-Esseen bound in a general setting, extending to functions of dynamical systems and random walks.

Findings

Derived near-sharp dependence conditions for optimal convergence rates.

Applicable to functions of the doubling map and random walks on linear groups.

Results demonstrate broad applicability across different stochastic processes.

Abstract

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random variable. Given sufficiently moments, when do we have a rate of convergence of $n^{-1/2}$ in the uniform metric, in other words, when do we have the optimal Berry-Esseen bound? We study this question in a quite general framework and find the (almost) sharp dependence conditions. The result applies to many different processes and dynamical systems. As specific, prominent examples, we study functions of the doubling map 2x mod 1 and the left random walk on the general linear group.