Strong convergence of ESD for the generalized sample covariance matrices when p/n→0

Abstract

Let X=[X ij]p×n be a p× n random matrix whose entries are i.i.d real random variables satisfying the moment condition E{double-struck}X 114<∞. Let T be a p× p deterministic nonnegative definite matrix. It is assumed that the empirical distribution of the eigenvalues of T converges weakly to a probability distribution. We consider the renormalized sample covariance matrix H̃=√n/p(1/nT 1/2XX tT 1/2-T) in the case of p/n→ 0 as p, n→ ∞. We study the limiting spectral distribution of H in this paper. The limiting distribution is shown to be coincident with the case of a generalized Wigner matrix considered in. Bai and Zhang (2010). © 2012 Elsevier B.V..