No Eigenvalues Outside the Support of the Limiting Spectral Distribution of Large Dimensional noncentral Sample Covariance Matrices

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, ${\mathbb{R}}_n$ is a $p\times n$ non-random matrix and $ \bbT_{n} $ is a $p\times p$ non-random, nonnegative definite Hermitian matrix. The matrix $\bbB_n$ is referred to as the information-plus-noise type matrix, where ${\mathbb{R}}_n$ contains the information and $\bbT^{1/2}_n \bbX_n$ is the noise matrix with the covariance matrix $\bbT_{n} $. It is known that, as $n\to \infty$, if $p/n$ converges to a positive number, the empirical spectral distribution of $ \bbB_n $ converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of ${\mathbb{R}}_n$ and $ \bbT_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.

26 pages