No eigenvalues outside the support of the limiting spectral distribution of large dimensional noncentral sample covariance matrices

Abstract

Let ${\mathbf{B}}_n=n^{-1}({\mathbf{R}}_n+{\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n){({\mathbf{R}}_n+{\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n)}^{\ast }$, where ${\mathbf{X}}_n$ is a $p\times n$ matrix with independent standardized random variables, ${\mathbf{R}}_n$ is a $p\times n$ non-random matrix and ${\mathbf{T}}_n$ is a $p\times p$ non-random, nonnegative definite Hermitian matrix. The matrix ${\mathbf{B}}_n$ is referred to as the information-plus-noise type matrix, where ${\mathbf{R}}_n$ contains the information and ${\mathbf{T}}_n^{1∕2}{\mathbf{X}}_n$ is the noise matrix with the covariance matrix ${\mathbf{T}}_n$. It is known that, as $n\to \mathrm{\infty }$, if $p∕n$ converges to a positive number, the empirical spectral distribution of ${\mathbf{B}}_n$ converges almost surely to a nonrandom limit, under some conditions. In this paper, we prove that, under certain conditions on the eigenvalues of ${\mathbf{R}}_n$ and ${\mathbf{T}}_n$, 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 n sufficiently large.