Distribution of eigenvalues of sample covariance matrices with tensor product samples

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.

References in corpus (2)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices

Cited by in corpus (1)

• CLT for linear eigenvalue statistics for a tensor product version of sample covariance matrices