Convergence rates of spectral distributions of large sample covariance matrices. (English) Zbl 1059.60036

Bai, Z. D.; Miao, Baiqi; Yao, Jian-feng

SIAM J. Matrix Anal. Appl. 25, No. 1, 105-127 (2003).

Summary: We improve known results on the convergence rates of spectral distributions of large-dimensional sample covariance matrices of size \(p {\times} n\). Using the Stieltjes transform, we first prove that the expected spectral distribution converges to the limiting Marchenko-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 \(O_{\text{p}}(n^{-2/5})\) and \(o_{\text{a.s.}}(n^{-2/5+\eta})\), 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.

MSC:

60F15	Strong limit theorems
62H99	Multivariate analysis

Keywords:

convergence rate; random matrix; spectral distribution; Marchenko-Pastur distribution