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 Marčenko-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.