  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.
