Exact Separation of Eigenvalues of Large Dimensional Noncentral Sample Covariance Matrices

TL;DR

This paper investigates the spectral properties of large noncentral sample covariance matrices, establishing an exact separation phenomenon for eigenvalues outside the support of their limiting spectral distribution.

Contribution

It provides a rigorous analysis of eigenvalue separation for large noncentral covariance matrices, extending understanding of their spectral distribution support.

Findings

Eigenvalues outside the support are exactly separated with high probability.

The number of eigenvalues on either side of certain intervals is precisely determined.

The results hold under specific conditions on the matrices involved.

Abstract

Let $ \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* $ where $ \bbX_n $ is a $ p \times n $ matrix with independent standardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrix, representing the information, and $ \bbT_{n} $ is a $ p \times p $ non-random nonnegative definite Hermitian matrix. Under some conditions on $ \bbR_n \bbR_n^* $ and $ \bbT_n $, it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $ p $ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals.