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

Let B-n = (1/N)T-n(1/2) XnXn*T-n(1/2), where X-n is n x N with i.i.d, complex standardized entries having finite fourth moment and T(n)1/2 is a Hermitian square root of the nonnegative definite Hermitian matrix T-n. It is known that, as n --> infinity, if n/N converges to a positive number and the empirical distribution of the eigenvalues of T-n converges to a proper probability distribution, then the empirical distribution of the eigenvalues of B-n converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of T-n, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all n sufficiently large.