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July 1999 Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices
Z. D. Bai, Jack W. Silverstein
Ann. Probab. 27(3): 1536-1555 (July 1999). DOI: 10.1214/aop/1022677458


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 was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of $T_n$, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large $n$. For these $n$ the interval corresponds to one that separates the eigenvalues of $T_n$. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of $B_n$ and $T_n$ lying on one side of their respective intervals are identical for all large $n$.


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Z. D. Bai. Jack W. Silverstein. "Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices." Ann. Probab. 27 (3) 1536 - 1555, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0964.60041
MathSciNet: MR1733159
Digital Object Identifier: 10.1214/aop/1022677458

Primary: 15A52 , 60F15
Secondary: 62H99

Keywords: empirical distribution function of eigenvalues , Random matrix , Stielt-jes transform

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
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