Multidimensional Stein method and quantitative asymptotic independence

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 Pimentel [Ann. Probab. 50 (2022), pp. 1755–1780], 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 $\mathbb {Y}$ 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 $(Z, \mathbb {Y})$ where $Z\sim N(0, \sigma ^{2})$ is independent of $\mathbb {Y}$. 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 $\mathbb {D} ^{1,2}$.