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

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 ${\mathbf{S}}_n=\frac{1}{n}{\mathbf{B}}_n{\mathbf{X}}_n{\mathbf{X}}_n^{\ast }{\mathbf{B}}_n^{\ast }$, 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.

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• Central limit theorem for linear spectral statistics of general separable sample covariance matrices with applications