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May 2012 Random covariance matrices: Universality of local statistics of eigenvalues
Terence Tao, Van Vu
Ann. Probab. 40(3): 1285-1315 (May 2012). DOI: 10.1214/11-AOP648


We study the eigenvalues of the covariance matrix 1/nMM of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and having finite C0th moment for some sufficiently large constant C0.

The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549–572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions.

As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C0th moment rather than exponential decay.


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Terence Tao. Van Vu. "Random covariance matrices: Universality of local statistics of eigenvalues." Ann. Probab. 40 (3) 1285 - 1315, May 2012.


Published: May 2012
First available in Project Euclid: 4 May 2012

zbMATH: 1247.15036
MathSciNet: MR2962092
Digital Object Identifier: 10.1214/11-AOP648

Primary: 15B52 , 62J10

Keywords: Covariance matrices , Four moment theorem , Universality

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.40 • No. 3 • May 2012
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