On Stein’s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein’s method to “infinite-variate” normal approximation that is an infinitedimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein’s identity for abstract Wiener measures and solve the corresponding Stein’s 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. © 2011 Elsevier Inc. All rights reserved.