Multidimensional Stein method and quantitative asymptotic independence

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 distribution, 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 independent 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 $\mathbb{D} ^{1,2}$.