High dimensional regimes of non-stationary Gaussian correlated Wishart matrices

TL;DR

This paper investigates the asymptotic behavior of high-dimensional correlated Wishart matrices with non-stationary Gaussian entries, revealing two regimes of convergence and applications to spectral distribution analysis.

Contribution

It identifies and characterizes two distinct asymptotic regimes for non-stationary Gaussian Wishart matrices and extends convergence results to functional settings.

Findings

In the central regime, Wishart matrices converge to Gaussian orthogonal ensemble matrices.

In the non-central regime, convergence is to Rosenblatt-Wishart matrices.

Spectral distributions converge to the semicircular law in the central regime.

Abstract

We study the high-dimensional asymptotic regimes of correlated Wishart matrices $d^{-1}\mathcal{Y}\mathcal{Y}^T$, where $\mathcal{Y}$ is a $n\times d$ Gaussian random matrix with correlated and non-stationary entries. We prove that under different normalizations, two distinct regimes emerge as both $n$ and $d$ grow to infinity. The first regime is the one of central convergence, where the law of the properly renormalized Wishart matrices becomes close in Wasserstein distance to that of a Gaussian orthogonal ensemble matrix. In the second regime, a non-central convergence happens, and the law of the normalized Wishart matrices becomes close in Wasserstein distance to that of the so-called Rosenblatt-Wishart matrix recently introduced by Nourdin and Zheng. We then proceed to show that the convergences stated above also hold in a functional setting, namely as weak convergence in $C([a,b];M_n(\mathbb{R}))$. As an application of our main result (in the central convergence regime), we show that it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. Our findings complement and extend a rich collection of results on the study of the fluctuations of Gaussian Wishart matrices, and we provide explicit examples based on Gaussian entries given by normalized increments of a bi-fractional or a sub-fractional Brownian motion.