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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 independentstandardized random variables, $ \bbR_n $ is a $ p \times n $ non-random matrixand $ \bbT_{n} $ is a $ p \times p $ non-random, nonnegative definite Hermitianmatrix. The matrix $\bbB_n$ is referred to as the information-plus-noise typematrix, where $\bbR_n$ contains the information and $\bbT^{1/2}_n \bbX_n$ isthe 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 empiricalspectral distribution of $ \bbB_n $ converges almost surely to a nonrandomlimit, under some mild conditions. In this paper, we prove that, under certainconditions on the eigenvalues of $ \bbR_n $ and $ \bbT_n $, for any closedinterval outside the support of the limit spectral distribution, withprobability one there will be no eigenvalues falling in this interval for all $n $ sufficiently large.

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