Total variation estimates in the Breuer-Major theorem

TL;DR

This paper establishes new bounds for the convergence rate in total variation distance within the Breuer-Major theorem, leveraging Malliavin calculus and Stein's method, with applications to fractional Brownian motion analysis.

Contribution

It introduces novel bounds for total variation convergence rates in the Breuer-Major theorem using a divergence representation and Hermite polynomial shifts.

Findings

Derived bounds for total variation distance between divergence and Gaussian variables

Applied results to power variations of fractional Brownian motion

Provided methods for Hurst parameter estimation

Abstract

This paper provides estimates for the convergence rate of the total variation distance in the framework of the Breuer-Major theorem, assuming some smoothness properties of the underlying function. The results are proved by applying new bounds for the total variation distance between a random variable expressed as a divergence and a standard Gaussian random variable, which are derived by a combination of techniques of Malliavin calculus and Stein's method. The representation of a functional of a Gaussian sequence as a divergence is established by introducing a shift operator on the expansion in Hermite polynomials. Some applications to the asymptotic behavior of power variations of the fractional Brownian motions and to the estimation of the Hurst parameter using power variations are presented.