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

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 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 $\bbR_n$ 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 $\bbR_n$ 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.