Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

Abstract

In this note, we study the Gaussian fluctuations for the Wishart matrices d −1 X n,d X T n,d , where X n,d is a n × d random matrix whose entries are jointly Gaussian and correlated with row and column covariance functions given by r and s respectively such that r(0) = s(0) = 1. Under the assumptions s ∈ ℓ 4/3 (Z) and r ℓ 1 (Z) < √ 6/2, we establish the n 3 /d convergence rate for the Wasserstein distance between a normalization of d −1 X n,d X T n,d and the corresponding Gaussian ensemble. This rate is the same as the optimal one computed in [3][4][5] for the total variation distance, in the particular case where the Gaussian entries of X n,d are independent. Similarly, we obtain the n 2p−1 /d convergence rate for the Wasserstein distance in the setting of random p-tensors of overall correlation. Our analysis is based on the Malliavin-Stein approach.