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January 1998 No eigenvalues outside the support of the limiting spectraldistribution of large-dimensional sample covariance matrices
Z. D. Bai, Jack W. Silverstein
Ann. Probab. 26(1): 316-345 (January 1998). DOI: 10.1214/aop/1022855421

Abstract

Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.

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Z. D. Bai. Jack W. Silverstein. "No eigenvalues outside the support of the limiting spectraldistribution of large-dimensional sample covariance matrices." Ann. Probab. 26 (1) 316 - 345, January 1998. https://doi.org/10.1214/aop/1022855421

Information

Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60017
MathSciNet: MR1617051
Digital Object Identifier: 10.1214/aop/1022855421

Subjects:
Primary: 60F15
Secondary: 62H99

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.26 • No. 1 • January 1998
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