A Note on the Convergence Rate of the Spectral Distributions of Large Sample Covariance Matrices

In this note, we improve the convergence rates of spectral distributions that large-dimensional sample covariance matrices of size p X n tend to the Mareenko-Pastur distribution, under the assumption that the entries have a finite eighth moment. Especially, we show that the expected spectral distributions of large-dimensional sample covariance matrices of size p X n tends to the limiting distribution with the dimension sample size ratio y yn p/n at the rate of 0(n-1/6) if y is close to 1 . Similar results for both the convergence in probability and the almost sure convergence are shown to be Op (n-1/6) and Oa,s (n-1/6), under the condition that y is close to 1.