The optimal fourth moment theorem

TL;DR

This paper precisely quantifies the convergence rates in total variation for the fourth moment theorem, linking the CLT in Wiener chaos to the decay of fourth cumulants, with an illustrative example.

Contribution

It provides exact convergence rates in total variation for the fourth moment theorem and illustrates these results with a Gaussian-subordinated sequence example.

Findings

Exact rates of convergence in total variation are derived.

The fourth cumulant convergence characterizes the CLT in Wiener chaos.

An explicit example based on the Breuer-Major CLT is provided.

Abstract

We compute the exact rates of convergence in total variation associated with the 'fourth moment theorem' by Nualart and Peccati (2005), stating that a sequence of random variables living in a fixed Wiener chaos verifies a central limit theorem (CLT) if and only if the sequence of the corresponding fourth cumulants converges to zero. We also provide an explicit illustration based on the Breuer-Major CLT for Gaussian-subordinated random sequences.