Asymptotic Behavior of Large Gaussian Correlated Wishart Matrices

Abstract

We consider high-dimensional Wishart matrices \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\), associated with a rectangular random matrix \({\mathcal {X}}_{n,d}\) of size \(n\times d\) whose entries are jointly Gaussian and correlated. Even if we will consider the case of overall correlation among the entries of \({\mathcal {X}}_{n,d}\), our main focus is on the case where the rows of \({\mathcal {X}}_{n,d}\) are independent copies of an n-dimensional stationary centered Gaussian vector of correlation function s. When s belongs to \(\ell ^{4/3}({\mathbb {Z}})\), we show that a proper normalization of \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\) is close in Wasserstein distance to the corresponding Gaussian ensemble as long as d is much larger than \(n^3\), thus recovering the main finding of Bubeck et al. (Random Struct Algorithms 49(3):503–532, 2016) and Jiang and Li (J Theor Probab 28(3):804–847, 2015) and extending it to a larger class of matrices. We also investigate the case where s is the correlation function associated with the fractional Brownian noise of parameter H. This example is very rich, as it gives rise to a great variety of phenomena with very different natures, depending on how H is located with respect to 1/2, 5/8 and 3/4. Notably, when \(H>3/4\), our study highlights a new probabilistic object, which we have decided to call the Rosenblatt–Wishart matrix. Our approach crucially relies on the fact that the entries of the Wishart matrices we are dealing with are double Wiener–Itô integrals, allowing us to make use of multivariate bounds arising from the Malliavin–Stein method and related ideas. To conclude the paper, we analyze the situation where the row-independence assumption is relaxed and we also look at the setting of random p-tensors (\(p\ge 3\)), a natural extension of Wishart matrices.