Convergence Rates of Spectral Distributions of Large Sample Covariance Matrices

Abstract

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=yn=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 oa.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.