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

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

2005-03-07

arxiv.org/abs/math/0503109v2

Annals of Probability 2007, Vol. 35, No. 2, 663-714

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000917 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000917

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 $n\times p$ matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance $\Sigma_p$. We show that for a large class of covariance matrices $\Sigma_p$, 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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