A note on a local limit theorem for Wiener space valued random variables

TL;DR

This paper establishes a local limit theorem for i.i.d. random variables on an abstract Wiener space, extending classical CLT results to an infinite-dimensional Gaussian setting with explicit density convergence conditions.

Contribution

It introduces the Ornstein-Uhlenbeck semigroup and Wick product as key tools for proving a local limit theorem in infinite-dimensional Gaussian spaces, providing new density convergence criteria.

Findings

Proves L1-convergence of densities for normalized sums of Wiener space valued variables.

Identifies necessary and sufficient conditions on densities for the local limit theorem to hold.

Compares the result with Berry-Esseen bounds in multidimensional CLTs.

Abstract

We prove a local limit theorem, i.e. a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional Gaussian framework are played by the Ornstein-Uhlenbeck semigroup and Wick product, respectively. We proceed by establishing a necessary condition on the density of the random variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional assumption the desired L1-convergence of the density of \frac{X_1+...+X_n}{\sqrt{n}}. We close the paper comparing our result with certain Berry-Esseen bounds for multidimensional central limit theorems.