Multidimensional Stein method and quantitative asymptotic independence

TL;DR

This paper develops a multidimensional Stein-Malliavin calculus to measure the distance between joint distributions and proves asymptotic independence in Wiener chaos sequences, with broad applications in probability theory.

Contribution

It introduces a new multidimensional Stein-Malliavin calculus framework and establishes joint convergence and asymptotic independence results for Wiener chaos sequences.

Findings

Established bounds for Wasserstein distance using Malliavin operators.

Proved joint convergence to independent Gaussian and arbitrary vectors.

Demonstrated automatic satisfaction of assumptions in specific Wiener chaos cases.

Abstract

If $\mathbb{Y}$ is a random vector in $\mathbb{R}^{d}$, we denote by $P_{\mathbb{Y}}$ its probability distribution. Consider a random variable $X$ and a $d$-dimensional random vector $\mathbb{Y}$. Inspired by \cite{Pi}, we develop a multidimensional Stein-Malliavin calculus which allows to measure the Wasserstein distance between the law $P_{ (X, \mathbb{Y})}$ and the probability distribution $P_{Z}\otimes P_{\mathbb{Y}}$, where $Z$ is a Gaussian random variable. That is, we give estimates, in terms of the Malliavin operators, for the distance between the law of the random vector $(X, \mathbb{Y})$ and the law of the vector $(Z, \mathbb{Y})$, where $Z$ is Gaussian and independent of $\mathbb{Y}$. Then we focus on the particular case of random vectors in Wiener chaos and we give an asymptotic version of this result. In this situation, this variant of the Stein-Malliavin calculus has strong and unexpected consequences. Let $(X_{k}, k\geq 1)$ be a sequence of random variables in the $p$th Wiener chaos ($p\geq 2$), which converges in law, as $k\to \infty$, to the Gaussian distribution $N(0, \sigma^2)$. Also consider $(\mathbb{Y}_{k}, k\geq 1)$ a $d$-dimensional random sequence converging in $L^{2}(\Omega)$, as $k\to \infty$, to an arbitrary random vector $\mathbb{U}$ in $\mathbb{R}^{d}$ and assume that the two sequences are asymptotically uncorrelated. We prove that, under very light assumptions on $\mathbb{Y}_{k}$, we have the joint convergence of $(X_{k}, \mathbb{Y}_{k}), k\geq 1)$ to $(Z, \mathbb{U})$ where $Z\sim N(0, \sigma ^{2})$ is indeendent of $\mathbb{U}$. These assumptions are automatically satisfied when the components of the vector $\mathbb{Y}_{k}$ belong to a finite sum of Wiener chaoses or when $\mathbb{Y}_{k}=Y$ for every $k\geq 1$, where $\mathbb{Y}$ belongs to the Sobolev-Malliavin space $\mathbb{D}^{1,2}$.