Hadamard product of independent random sample covariance matrices with correlation structure

TL;DR

This paper derives the limiting eigenvalue distribution of a Hadamard product of scaled sample covariance matrices with correlated rows, under high-dimensional asymptotics, extending understanding of spectral properties in complex covariance structures.

Contribution

It provides the first explicit characterization of the asymptotic eigenvalue distribution for Hadamard products of correlated sample covariance matrices.

Findings

Derived the asymptotic eigenvalue distribution for the matrix M.

Extended random matrix theory to matrices with correlated rows.

Applicable to high-dimensional covariance analysis with complex correlation structures.

Abstract

We compute the asymptotic empirical eigenvalue distribution of the matrix $M = \bigodot_{i=1}^k \frac{1}{d_i}X^{(i)}{X^{(i)}}^\top$ where $X^{(i)}\in\mathbb{R}^{n\times d_i}$ are independent matrices with independent rows but general correlation within each row under the dimension scaling $\frac{n}{d_1\dots d_k}\to \gamma$.