Multivariate normal approximation using Stein's method and Malliavin calculus

TL;DR

This paper integrates Stein's method with Malliavin calculus to derive explicit bounds for the multidimensional normal approximation of Gaussian field functionals, extending previous results and applying to complex covariance structures.

Contribution

It introduces a new approach combining Stein's method and Malliavin calculus for explicit bounds in Gaussian approximation, applicable to arbitrary covariance matrices.

Findings

Generalizes previous multidimensional normal approximation results

Provides explicit bounds in Wasserstein distance

Applies to fractional Brownian motion fields

Abstract

We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.