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 , where X_n is an n×N matrix consisting of i.i.d. standardized complex entries, 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 , 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.