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 Cn=(1/N)An1/2XnBnXn∗×An1/2, where Xn is an n×N matrix consisting of i.i.d. standardized complex entries, An1/2 is a nonnegative definite square root of the nonnegative definite Hermitian matrix An, and Bn is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of An and Bn converge to proper probability distributions as nN→c∈(0,∞), the empirical spectral distribution of Cn converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of An and Bn, 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.