Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

In this note, we study the Gaussian fluctuations for Wishart matrices d−1Xn,dXn,dT, where Xn,d is a n×d random matrix whose entries are jointly and correlated with row column covariance functions given by r s respectively such that r(0)=s(0)=1. Under assumptions s∈ℓ4/3(Z) ‖r‖ℓ1(Z)<6/2, establish n3/d convergence rate Wasserstein distance between normalization of d−1Xn,dXn,dT corresponding ensemble. This same as optimal one computed in Bubeck et al. (2016), Ganguly (2018) Jiang Li (2015) total variation distance, particular case independent. Similarly, obtain n2p−1/d setting symmetric tensor order p overall correlation. Our analysis based on Malliavin–Stein approach.