CLT for linear spectral statistics of large-dimensional sample covariance matrices.
type: Item
instance of: scholarly article
title: CLT for linear spectral statistics of large-dimensional sample covariance matrices.
zbMATH Open document ID: 1063.60022
author: Jack W. Silverstein
author: Zhidong Bai
published in: The Annals of Probability
publication date: 2004-09-15T00:00:00Z
review text: Statisticians are confronted with the task of analyzing data with high-dimensional data. In the past, so-called dimension reduction schemes were used in dealing with it. But even it is possible to compute much of what is needed, there is a fundamental problem with the analytical tools used by statisticians. Their use relies on their asymptotic behavior as the number of samples increase. Some methods behave very poorly and some are even not applicable. Let \(B_n = (1/N)T^{1/2}_{n}X_{n}X^{*}_{n}T^{1/2}_{n}\) where \(X_n =(X_{ij})\) is \(n \times N\) with i.i.d. complex standardized entries having finite fourth moment, and \(T^{1/2}_{n}\) is a Hermitian square root of the nonnegative definite Hermitian matrix \(T_{n}\). The limiting behavior, as \(n \rightarrow\infty\) with \(n/N\) approaching a positive constant, of functionals of the eigenvalues of \(B_n\), where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of \(B_n\), it is known that these linear spectral statistics converge a.s. to a nonrandom quantity. The authors show their rate of convergence to be \(1/n\) by proving, after proper scaling, that they form a tight sequence. Moreover, they show if \(XX^{2}_{11}=0\) and \(X| X_{11}| ^4 =2\), or if \(X_{11}\) and \(T_n\) are real and \(XX^{4}_{11} =3\), they are shown to have Gaussian limits. The main theorem can be viewed as an extension of results obtained by \textit{D. Jonsson} [J. Multivariate Anal. 12, 1-38 (1982; Zbl 0491.62021)] where the entries of \(X_n\) are Gaussian and \(T_n = I \) and is consistent with central limit theorem results on linear statistics of eigenvalues of other classes of random matrices. The techniques and arguments used to prove the theorem have nothing in common with any of the tools used in other papers.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 62H05
zbMATH DE Number: 2100728
zbMATH Keywords: linear spectral statistics
zbMATH Keywords: random matrix
zbMATH Keywords: empirical distribution function of eigenvalues
zbMATH Keywords: Stieltjes transform
reviewed by: Tae Il Jeon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1078415845
OpenAlex ID: W1965748679
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DOI: 10.1214/AOP/1078415845
description: scientific article; zbMATH DE number 2100728