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

Abstract

Let Bn =(1/N)T1/2 entries having finite fourthmoment, and T1/2 n XnX∗ n where Xn is n×N with i.i.d. complex standardized n is a Hermitian square root of the nonnegative nT1/2 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.