Multidimensional Stein–Malliavin calculus for the multivariate Gaussian distribution

We develop a Stein–Malliavin calculus in order to evaluate the Wasserstein distance between the probability law of a d+p dimensional random vector (X,Y), with components regular enough in the sense of Malliavin calculus, and the probability law of the vector (Z,Y), where Z is a centered d-dimensional Gaussian vector independent of Y. We apply our theory to quantify the asymptotic independence for sequences of random vectors. We focus on some particular examples that concern the random matrices and the parameter estimators for the stochastic wave equation.