Optimal local central limit theorems on Wiener chaos
Masahisa Ebina, Ivan Nourdin, Giovanni Peccati
TL;DR
The paper studies local central limit theorems for normalized Wiener-chaos functionals converging to $\mathcal{N}(0,1)$, proving that the optimal density-convergence rate in Sobolev spaces $W^{k,r}(\mathbb{R})$ is governed by $\mathbf{M}(F_n)=\max\{|\mathbb{E}[F_n^3]|,\mathbb{E}[F_n^4]-3\}$. The authors develop a novel tempered-distribution Stein framework, solving Stein's equation in $\mathcal{S}'(\mathbb{R})$ and using composition with delta distributions to obtain uniform control of density derivatives. They connect cumulants via Gamma-operators to Edgeworth-type expansions, establishing sharp rates in $L^r$ and $W^{k,r}$ norms, and they derive exact asymptotics under a two-dimensional convergence assumption, with applications to Breuer–Major-type functionals. The results enhance the Malliavin–Stein method by enabling direct density-level analysis, offering precise, optimal rates and asymptotics that have potential impact on a broad class of Gaussian functionals. Overall, the work provides a rigorous, unified approach to density convergence in Wiener chaos and clarifies the role of third and fourth cumulants in local CLTs across Sobolev scales.
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.