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

Let Bₙ = (1/N)Tₙ1/2Xₙ Xₙ^* Tₙ1/2, where Xₙ is n × N with i.i.d. complex standardized entries having finite fourth moment and Tₙ1/2 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.