Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

Abstract

In this note, we study the Gaussian fluctuations for the Wishart matrices d−1Xn,dXn,dT, where Xn,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∈ℓ4/3(Z) and ‖r‖ℓ1(Z)<6/2, we establish the n3/d convergence rate for the Wasserstein distance between a normalization of d−1Xn,dXn,dT and the corresponding Gaussian ensemble. This rate is the same as the optimal one computed in Bubeck et al. (2016), Bubeck and Ganguly (2018) and Jiang and Li (2015) for the total variation distance, in the particular case where the Gaussian entries of Xn,d are independent. Similarly, we obtain the n2p−1/d convergence rate for the Wasserstein distance in the setting of symmetric random tensor of order p with overall correlation. Our analysis is based on the Malliavin–Stein approach.