Exact separation of eigenvalues of large dimensional sample covariance matrices

Abstract

Let Bn = (1/N)T1/2 nXnX* nT1/2 n where Xn is n × N with i.i.d. complex standardized entries having finite fourth moment, and T1/2 n is a Hermitian square root of the nonnegative definite Hermitian matrix Tn. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of Tn, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large n. For these n the interval corresponds to one that separates the eigenvalues of Tn. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of Bn and Tn lying on one side of their respective intervals are identical for all large n.