Multivariate normal approximation on the Wiener space: new bounds in the convex distance

TL;DR

This paper provides explicit bounds on the convex distance between distributions of Gaussian functionals and normal vectors, improving previous bounds and applying to multivariate limit theorems in Wiener space.

Contribution

It introduces new bounds for the convex distance in Gaussian approximation, extending previous results and applying recursive estimates to multivariate and functional limit theorems.

Findings

Bounds are comparable to Wasserstein bounds but without logarithmic factors.

A multivariate fourth moment theorem for Wiener-Itô integrals is established.

Rates of convergence in the functional Breuer-Major theorem are characterized.

Abstract

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin, Peccati and R\'eveillac (2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener-It\^o integrals, and (ii) we characterise the rate of convergence for the finite-dimensional distributions in the functional Breuer-Major theorem.