The Annals of Probability

Random covariance matrices: Universality of local statistics of eigenvalues

Terence Tao and Van Vu

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Abstract

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.

Article information

Source
Ann. Probab., Volume 40, Number 3 (2012), 1285-1315.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1336136064

Digital Object Identifier
doi:10.1214/11-AOP648

Mathematical Reviews number (MathSciNet)
MR2962092

Zentralblatt MATH identifier
1247.15036

Subjects
Primary: 15B52: Random matrices 62J10: Analysis of variance and covariance

Keywords
Four moment theorem universality covariance matrices

Citation

Tao, Terence; Vu, Van. Random covariance matrices: Universality of local statistics of eigenvalues. Ann. Probab. 40 (2012), no. 3, 1285--1315. doi:10.1214/11-AOP648. https://projecteuclid.org/euclid.aop/1336136064


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