Multidimensional Stein method and quantitative asymptotic independence
Ciprian A. Tudor · Transactions of the American Mathematical Society · 2024
If Y \mathbb {Y} is a random vector in R d \mathbb {R} ^{d} , we denote by P Y P_{\mathbb {Y}} its probability distribution. Consider a random variable X X and a d d -dimensional random vector Y \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 , Y ) P_{ (X, \mathbb {Y})} and the probability distribution P Z ⊗ P Y P_{Z}\otimes P_{ \mathbb {Y}} , where Z 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 , Y ) (X, \mathbb {Y}) and the law of the vector ( Z , Y ) (Z, \mathbb {Y}) , where Z Z