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Title: Convergence rates of spectral distributions of large sample covariance matrices
Authors: Bai, Z.D. 
Miao, B.
Yao, J.-F.
Keywords: Convergence rate
Marčenko-Pastur distribution
Random matrix
Spectral distribution
Issue Date: 2004
Citation: Bai, Z.D., Miao, B., Yao, J.-F. (2004). Convergence rates of spectral distributions of large sample covariance matrices. SIAM Journal on Matrix Analysis and Applications 25 (1) : 105-127. ScholarBank@NUS Repository.
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.
Source Title: SIAM Journal on Matrix Analysis and Applications
ISSN: 08954798
DOI: 10.1137/S0895479801385116
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