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 × 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 O p (n -2/5 ) and o a.s. (n -2/5+η ), 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.