Asymptotic behavior of large Gaussian correlated Wishart matrices

TL;DR

This paper investigates the asymptotic behavior of large Gaussian correlated Wishart matrices, extending previous results to broader classes of correlation structures and exploring new phenomena related to fractional Brownian noise.

Contribution

It extends the understanding of Wishart matrices with correlated Gaussian entries, including cases with fractional Brownian noise, and introduces the Rosenblatt-Wishart matrix for certain parameters.

Findings

Proper normalization leads to Gaussian approximation when $d o ^3$

Different behaviors depending on the Hurst parameter $H$

Introduction of the Rosenblatt-Wishart matrix for $H>3/4$

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 a $n$-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 [3,9] 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-It\^o 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 ($p\geq 3$), a natural extension of Wishart matrices.