Convergence in total variation on Wiener chaos

TL;DR

This paper proves that sequences of Wiener chaos random variables converging in distribution also converge in total variation, providing bounds and extending classical theorems to stronger modes of convergence.

Contribution

It establishes total variation convergence for Wiener chaos sequences, extends Peccati-Tudor theorem, and provides bounds on the total variation distance.

Findings

Sequences in Wiener chaos converge in total variation under certain conditions.

The law of the limit is absolutely continuous.

Provides bounds on total variation distance for multiple Wiener-Itô integrals.

Abstract

Let ${F_n}$ be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards $F_\infty$ satisfying ${\rm Var}(F_\infty)>0$. Our first result is a sequential version of a theorem by Shigekawa (1980). More precisely, we prove, without additional assumptions, that the sequence ${F_n}$ actually converges in total variation and that the law of $F_\infty$ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each $F_n$ has more specifically the form of a multiple Wiener-It\^o integral (of a fixed order) and that it converges in $L^2(\Omega)$ towards $F_\infty$. We then give an upper bound for the distance in total variation between the laws of $F_n$ and $F_\infty$. As such, we recover an inequality due to Davydov and Martynova (1987); our rate is weaker compared to Davydov and Martynova (1987) (by a power of 1/2), but the advantage is that our proof is not only sketched as in Davydov and Martynova (1987). Finally, in a third part we show that the convergence in the celebrated Peccati-Tudor theorem actually holds in the total variation topology.