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

Let X=[Xij]p×nX=[Xij]p×n be a p×np×n random matrix whose entries are i.i.d real random variables satisfying the moment condition EX114<∞. Let TT be a p×pp×p deterministic nonnegative definite matrix. It is assumed that the empirical distribution of the eigenvalues of TT converges weakly to a probability distribution. We consider the renormalized sample covariance matrix H̃=np(1nT1/2XXtT1/2−T) in the case of p/n→0p/n→0 as p,n→∞p,n→∞. We study the limiting spectral distribution of H̃ 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).