Central limit theorem and convergence of the support for Wishart matrices with correlated entries

TL;DR

This paper extends the understanding of Wishart matrices with correlated entries by establishing a central limit theorem for linear statistics and analyzing the convergence of the support to the limiting measure.

Contribution

It introduces conditions under which the CLT holds for correlated Wishart matrices with log-concave laws and demonstrates support convergence.

Findings

CLT for linear statistics of correlated Wishart matrices

Support of empirical measures converges to the limit

Convergence of expectation and covariance

Abstract

In this article, we will consider Wishart Matrices with correlated entries, but with a strictly log-concave law. It has been shown by A.Pajor and L.Pastur that the empirical measures of such matrices converges. We will show, under some symmetry and convergence hypotheses, that in this case the central limit theorem for linear statistics holds around their limit, and deduce the convergence of the support to the support of the limiting measure. We also provide the convergence of the Expectation and the Covariance.