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March 2007 Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices
Noureddine El Karoui
Ann. Probab. 35(2): 663-714 (March 2007). DOI: 10.1214/009117906000000917

Abstract

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n×p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance Σp. We show that for a large class of covariance matrices Σp, the largest eigenvalue of X*X is asymptotically distributed (after recentering and rescaling) as the Tracy–Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p.

The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

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Noureddine El Karoui. "Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices." Ann. Probab. 35 (2) 663 - 714, March 2007. https://doi.org/10.1214/009117906000000917

Information

Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1117.60020
MathSciNet: MR2308592
Digital Object Identifier: 10.1214/009117906000000917

Subjects:
Primary: 60F05
Secondary: 62E20

Keywords: operator determinants , Random matrix theory , steepest descent analysis , Toeplitz matrices , trace class operators , Tracy–Widom distributions , Wishart matrices

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 2 • March 2007
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