Berry-Esseen bounds in the Breuer-Major CLT and Gebelein's inequality

TL;DR

This paper establishes explicit Berry-Esseen bounds in total variation distance for the Breuer-Major CLT, utilizing Malliavin-Stein methods and Gebelein's inequality to improve understanding of Gaussian functional approximations.

Contribution

It introduces a novel approach combining Malliavin-Stein techniques with Gebelein's inequality to derive explicit bounds in the Breuer-Major CLT under minimal regularity.

Findings

Explicit Berry-Esseen bounds in total variation distance.

Effective bounds for Gaussian functionals with minimal regularity.

Enhanced understanding of covariance bounds via Gebelein's inequality.

Abstract

We derive explicit Berry-Esseen bounds in the total variation distance for the Breuer-Major central limit theorem, in the case of a subordinating function $\varphi$ satisfying minimal regularity assumptions. Our approach is based on the combination of the Malliavin-Stein approach for normal approximations with Gebelein's inequality, bounding the covariance of functionals of Gaussian fields in terms of maximal correlation coefficients.