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

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