Convergence rates of spectral distributions of large sample covariance matrices

In this paper, we improve known results on the convergence rates of spectral distributions of large-dimensional sample covariance matrices of size p x n. Using the Stieltjes transform, we first prove that the expected spectral distribution converges to the limiting Marcenko-Pastur distribution with the dimension sample size ratio y = y(n) = p/n at a rate of O(n(-1/2)) if y keeps away from 0 and 1, under the assumption that the entries have a finite eighth moment. Furthermore, the rates for both the convergence in probability and the almost sure convergence are shown to be Op(n(-2/5)) and o(a.s.)(n(-2/5+eta)), respectively, when y is away from 1. It is interesting that the rate in all senses is O(n(-1/8)) when y is close to 1.