Distribution of eigenvalues of sample covariance matrices with tensor product samples

TL;DR

This paper analyzes the eigenvalue distribution of large sample covariance matrices formed from tensor product samples, establishing convergence to a deterministic limit characterized by a functional equation.

Contribution

It introduces a new framework for understanding eigenvalue distributions of tensor product-based sample covariance matrices with convergence results.

Findings

Eigenvalue distribution converges weakly to a deterministic limit.

The limiting spectral distribution is characterized by a specific functional equation.

Results hold under general conditions on the matrix B and the tensor product structure.

Abstract

We consider $n^2\times n^2$ real symmetric and hermitian matrices $M_n$, which are equal to sum of $m_n$ tensor products of vectors $X^\mu=B(Y^\mu\otimes Y^\mu)$, $\mu=1,\dots,m_n$, where $Y^\mu$ are i.i.d. random vectors from $\mathbb R^n (\mathbb C^n)$ with zero mean and unit variance of components, and $B$ is an $n^2\times n^2$ positive definite non-random matrix. We prove that if $m_n/n^2\to c\in [0,+\infty)$ and the Normalized Counting Measure of eigenvalues of $BJB$, where $J$ is defined below, converges weakly, then the Normalized Counting Measure of eigenvalues of $M_n$ converges weakly in probability to a non-random limit and its Stieltjes transform can be found from a certain functional equation.