No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix

We consider a class of matrices of the form C n = ( 1 / N ) A n 1 / 2 X n B n X n ∗ × A n 1 / 2 , where X n is an n × N matrix consisting of i.i.d. standardized complex entries, A n 1 / 2 is a nonnegative definite square root of the nonnegative definite Hermitian matrix A n , and B n is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of A n and B n converge to proper probability distributions as n N → c ∈ ( 0 , ∞ ) , the empirical spectral distribution of C n converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of A n and B n , with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n . The problem is motivated by applications in spatio-temporal statistics and wireless communications.