Convergence rates to the Marchenko-Pastur type distribution

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Bai, Zhidong^1,2,3  Hu, Jiang^1,2  Zhou, Wang³ 

[1] NE Normal Univ, KLASMOE, Changchun 130024, Peoples R China

[2] NE Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China

[3] Natl Univ Singapore, Dept Stat & Appl Probabil, Singapore 117546, Singapore

关键词: Convergence rate;    Sample covariance matrix;    Spectral distribution;   

DOI  :  10.1016/j.spa.2011.10.002

来源: Elsevier

【 摘 要 】

S-n = 1/n (TnXnX)-X-1/2*T-n(n)1/2, where X-n = (x(ij)) is a p x n matrix consisting of independent complex entries with mean zero and variance one, T-n is a p x p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in p. In this paper, if sup(n) sup(i,j) E vertical bar x(ij)(8) vertical bar < infinity and y(n) = p/n < 1 uniformly as n -> infinity, we obtain that the rate of the expected empirical spectral distribution of S-n converging to its limit spectral distribution is O(n(-1/2)). Moreover, under the same assumption, we prove that for any eta > 0, the rates of the convergence of the empirical spectral distribution of S-n in probability and the almost sure convergence are O(n(-2/5)) and O(n(-2/5+eta)) respectively. (C) 2011 Elsevier B.V. All rights reserved.

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