Joint convergence of sample cross-covariance matrices

Article Type

Research Article

Publication Title

Alea (Rio de Janeiro)

Abstract

Suppose X and Y are p × n matrices each with entries that have mean 0, variance 1, and which have all moments of any order that are uniformly bounded as p; n → ∞. Moreover, the entries (Xij; Yij) are independent across i,j with a common correlation ρ. Let C = n-1XY* be the sample cross-covariance matrix. We show that if n, p → 1; p/n → y 6≠0, then C converges in the algebraic sense and the limit moments depend only on ρ. Independent copies of such matrices with same p but different n, say fnlg, different correlations {ρ}, and different non-zero y’s, say fylg, also converge jointly, and are asymptotically free. When y = 0, the matrix (Formula Presented)converges to an elliptic variable with parameter ρ2. In particular, this elliptic variable is circular when ρ= 0, and is semi-circular when ρ = 1. If we take independent Cl, then the matrices (Formula Presented) converge jointly and are also asymptotically free. As a consequence, the limiting spectral distribution of any symmetric matrix which is a polynomial in the above scaled and centered matrices exists and has compact support.

First Page

395

Last Page

423

DOI

https://10.30757/ALEA.V20-14

Publication Date

1-1-2023

Comments

Open Access, Bronze, Green

This document is currently not available here.

Share

COinS