Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants

TL;DR

This paper establishes precise bounds on the rate of convergence in the Central Limit Theorem for sequences in Wiener chaos, highlighting the roles of third and fourth cumulants, especially in the context of fractional Gaussian noise.

Contribution

It provides the first explicit bounds linking convergence rates to third and fourth cumulants for chaotic variables, extending the understanding of CLT rates on Wiener space.

Findings

Bounds depend on third and fourth cumulants

Optimal convergence rates are characterized for fractional Gaussian noise

Results apply to Breuer-Major CLT with sharp rates

Abstract

Let {F_n} be a normalized sequence of random variables in some fixed Wiener chaos associated with a general Gaussian field, and assume that E[F_n^4] --> E[N^4]=3, where N is a standard Gaussian random variable. Our main result is the following general bound: there exist two finite constants c,C>0 such that, for n sufficiently large, c max(|E[F_n^3]|, E[F_n^4]-3) < d(F_n,N) < C max(|E[F_n^3]|, E[F_n^4]-3), where d(F_n,N) = sup |E[h(F_n)] - E[h(N)]|, and h runs over the class of all real functions with a second derivative bounded by 1. This shows that the deterministic sequence max(|E[F_n^3]|, E[F_n^4]-3) completely characterizes the rate of convergence (with respect to smooth distances) in CLTs involving chaotic random variables. These results are used to determine optimal rates of convergence in the Breuer-Major central limit theorem, with specific emphasis on fractional Gaussian noise.