No eigenvalues outside the support of the limiting spectral distribution of large dimensional noncentral sample covariance matrices

Let B n = n - 1 ( R n + T 1 / 2 n X n )( R n + T 1 / 2 n Xn)*, where Xn is a p x n matrix with independent standardized random variables, Rn is a p x n non-random matrix and Tn is a p x p non-random, nonnegative definite Hermitian matrix. The matrix Bn is referred to as the information-plus-noise type matrix, where Rn contains the information and T 1 / 2 n Xn is the noise matrix with the covariance matrix Tn. It is known that, as n- OO , if p/n converges to a positive number, the empirical spectral distribution of Bn converges almost surely to a nonrandom limit, under some conditions. In this paper, we prove that, under certain conditions on the eigenvalues of Rn and Tn, 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.