The Annals of Probability

Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices

Noureddine El Karoui

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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.

Article information

Source
Ann. Probab., Volume 35, Number 2 (2007), 663-714.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1175287758

Digital Object Identifier
doi:10.1214/009117906000000917

Mathematical Reviews number (MathSciNet)
MR2308592

Zentralblatt MATH identifier
1117.60020

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory

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

Citation

El Karoui, Noureddine. Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35 (2007), no. 2, 663--714. doi:10.1214/009117906000000917. https://projecteuclid.org/euclid.aop/1175287758


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