Berry-esseen bounds in the breuer-major CLT and 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 ϕ 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.