No eigenvalues outside the limiting support of the spectral distribution of general sample covariance matrices

TL;DR

This paper proves that, under mild conditions, the eigenvalues of a broad class of sample covariance matrices almost surely do not appear outside their limiting spectral support, extending classical results like Bai-Yin law.

Contribution

It establishes the absence of eigenvalues outside the limiting spectral support for general sample covariance matrices, extending Bai-Yin law to more general settings.

Findings

Eigenvalues do not appear outside the limiting support with probability 1.

Extension of Bai-Yin law to broader matrix classes.

Results hold for matrices with arbitrary dimensions m.

Abstract

This paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form $\mathbf S_n=\frac1n\mathbf B_n\mathbf X_n\mathbf X_n^*\mathbf B_n^*$, where $\mathbf B_n$ is a $p\times m$ non-random matrix and $\mathbf X_n$ is an $m\times n$ matrix consisting of i.i.d standard complex entries. $p/n\to c\in (0,\infty)$ as $n\to \infty$ while $m$ can be arbitrary. We proved that under some mild assumptions, with probability 1, there will be no eigenvalues in any closed interval contained in an open interval outside the supports of the limiting distribution $F_{c_n,H_n}$, for all sufficiently large $n$. An extension of Bai-Yin law is also obtained.