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

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Abstract

Suppose that the (normalised) partial sum of a stationary sequence converges to a standard normal random vari-able. 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, the left random walk on the general linear group and functions of linear processes.

Original languageEnglish
Pages (from-to)1219-1245
Number of pages27
JournalBernoulli: a journal of mathematical statistics and probability
Volume29
Issue number2
Early online dateFeb 2023
DOIs
Publication statusPublished - May 2023

Austrian Fields of Science 2012

  • 101019 Stochastics

Keywords

  • Berry-Esseen
  • weak dependence

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