LSD of sample covariances of superposition of matrices with separable covariance structure

TL;DR

This paper analyzes the spectral distribution of matrices formed by superimposing multiple matrices with separable covariance structures, establishing their asymptotic behavior as dimensions grow large.

Contribution

It introduces a general framework for the LSD of superpositions of matrices with separable covariance, extending previous results to more complex matrix combinations.

Findings

Existence of a limiting spectral distribution for the superimposed matrices.

Characterization of the LSD via a system of equations with unique solutions.

Generalization of prior results on sample covariance matrices with separable structures.

Abstract

We study the asymptotic behavior of the spectra of matrices of the form $S_n = \frac{1}{n}XX^*$ where $X =\sum_{r=1}^K X_r$, where $X_r = A_r^\frac{1}{2}Z_rB_r^\frac{1}{2}$, $K \in \mathbb{N}$ and $A_r,B_r$ are sequences of positive semi-definite matrices of dimensions $p\times p$ and $n\times n$, respectively. We establish the existence of a limiting spectral distribution for $S_n$ by assuming that matrices $\{A_r\}_{r=1}^K$ are simultaneously diagonalizable and $\{B_r\}_{r=1}^K$ are simultaneously digaonalizable, and that the joint spectral distributions of $\{A_r\}_{r=1}^K$ and $\{B_r\}_{r=1}^K$ converge to $K$-dimensional distributions, as $p,n\to \infty$ such that $p/n \to c \in (0,\infty)$. The LSD of $S_n$ is characterized by system of equations with unique solutions within the class of Stieltjes transforms of measures on $\mathbb{R}_+$. These results generalize existing results on the LSD of sample covariances when the data matrices have a separable covariance structure.