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

Abstract

We prove a local limit theorem, that is, 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 $\mathcal{L}^{1}$-convergence of the density of $\frac{X_{1}+\cdots+X_{n}}{\sqrt{n}}$. We close the paper comparing our result with certain Berry–Esseen bounds for multidimensional central limit theorems.