On Steinʼs method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

Abstract

In this paper, we generalize Steinʼs method to “ infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Steinʼs identity for abstract Wiener measures and solve the corresponding Steinʼs equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg–Lévy type limit theorem. In addition, an analogous of Berry–Esséen type estimate for abstract Wiener measures will be obtained.