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

Let Bₙ = (1/N) Tₙ^1/2Xₙ Xₙ^* Tₙ^1/2, where Xₙ is n N with i. i. d. complex standardized entries having finite fourth moment and Tₙ^1/2 is a Hermitian square root of the nonnegative definite Hermitian matrix Tₙ. It is known that, as n, if n/N converges to a positive number and the empirical distribution of the eigenvalues of Tₙ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of Bₙ converges a. s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of Tₙ, 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.