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

Abstract

We study the asymptotic behavior of the spectra of matrices of the form $S_n=\frac{1}{n}X{X}^{\ast }$ where $X={\sum }_{r=1}^K{X}_r$, where $X_r=A_r^{\frac{1}{2}}{Z}_r{B}_r^{\frac{1}{2}}$, $K\in \mathbb{N}$ 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.