Exact separation of eigenvalues of large dimensional sample covariance matrices
type: Item
instance of: scholarly article
title: Exact separation of eigenvalues of large dimensional sample covariance matrices
zbMATH Open document ID: 0964.60041
author: Jack W. Silverstein
author: Zhidong Bai
published in: The Annals of Probability
publication date: 2000-11-22T00:00:00Z
review text: This paper is devoted to the study of the fine structure of the spectra of large random matrices \(B_n={1\over N}X_nX^*_n T_n\), where \(X_n= (X_{ij})\) is an \(n\times N\) matrix consisting of i.i.d. standardized complex random variables, \(T_n\) is an \(n\times n\) nonnegative definite nonrandom matrix; the limit \(n/N\to C>0\), \(n\to\infty\), is considered. It is assumed that the limiting eigenvalue distribution function (e.d.f.) \(F_T\) exists, and the \(T_n\) are bounded. It is known that under certain conditions on the distribution of \(X_{ij}\), the limiting e.d.f. \(F_B\) and \(F_{\underline B}\) exist, where \(\underline B_n={1\over n} X_n T_nX^*_n\). It is proved that if the \(T_n\) are such that there are eigenvalues \(\lambda^{(T_n)}_{i_n}\) and \(\lambda^{(T_{n+1})}_{i_{n+1}}\) situated at different sides of an interval \(Y\), then the same is true with probability \(1\) for the eigenvalues of \(B_n\), in the limit \(n\to\infty\). \(Y\) is determined in terms of \(F_{\underline B}\). This theorem improves the results by the same authors [ibid. 26, No. 1, 316-345 (1998; Zbl 0937.60017)].
reviewed by: A. Khorunzhy
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 15B52
zbMATH DE Number: 1462562
zbMATH Keywords: random matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: Wishart ensemble
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677458
OpenAlex ID: W1482572935
cites work: No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices
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DOI: 10.1214/AOP/1022677458
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description: scientific article; zbMATH DE number 1462562