Joint convergence of sample cross-covariance matrices

TL;DR

This paper studies the asymptotic behavior of sample cross-covariance matrices for large matrices with bounded moments, revealing their convergence properties, joint asymptotic freeness, and spectral distribution characteristics.

Contribution

It establishes the joint convergence and asymptotic freeness of sample cross-covariance matrices with correlated entries, extending random matrix theory results.

Findings

Sample cross-covariance matrices converge in the algebraic sense as dimensions grow.

Joint convergence and asymptotic freeness are proven for multiple such matrices with different parameters.

Limiting spectral distributions of polynomial functions of these matrices have compact support.

Abstract

Suppose $X$ and $Y$ are $p\times n$ matrices each with mean $0$, variance $1$ and where all moments of any order are uniformly bounded as $p,n \to \infty$. Moreover, the entries $(X_{ij}, Y_{ij})$ are independent across $i,j$ with a common correlation $\rho$. Let $C=n^{-1}XY^*$ be the sample cross-covariance matrix. We show that if $n, p\to \infty, p/n\to y\neq 0$, then $C$ converges in the algebraic sense and the limit moments depend only on $\rho$. Independent copies of such matrices with same $p$ but different $n$, say $\{n_l\}$, different correlations $\{\rho_l\}$, and different non-zero $y$'s, say $\{y_l\}$ also converge jointly and are asymptotically free. When $y=0$, the matrix $\sqrt{np^{-1}}(C-\rho I_p)$ converges to an elliptic variable with parameter $\rho^2$. In particular, this elliptic variable is circular when $\rho=0$ and is semi-circular when $\rho=1$. If we take independent $C_l$, then the matrices $\{\sqrt{n_lp^{-1}}(C_l-\rho_l I_p)\}$ converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix polynomial exists and has compact support.