Optimal local central limit theorems on Wiener chaos

Abstract

This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional conditions, that the optimal rate of convergence of their density functions to the standard normal density in the Sobolev space $W^{k,r}(\mathbb{R})$, for every $k \in \mathbb{N} \cup \{0\}$ and $r \in [1,\infty]$, is determined by the maximum of the absolute values of their third and fourth cumulants. We also obtain exact asymptotics for this convergence under an additional assumption. Our approach is based on Malliavin--Stein techniques combined with tools from the theory of generalized functionals in Malliavin calculus.