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We consider the asymptotic fluctuation behavior of the largest eigenvalue ofcertain sample covariance matrices in the asymptotic regime where bothdimensions 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 vectorswith mean 0 and covariance $\Sigma_p$. We show that for a large class ofcovariance matrices $\Sigma_p$, the largest eigenvalue of $X^*X$ isasymptotically distributed (after recentering and rescaling) as theTracy--Widom distribution that appears in the study of the Gaussian unitaryensemble. We give explicit formulas for the centering and scaling sequencesthat are easy to implement and involve only the spectral distribution of thepopulation covariance, $n$ and $p$. The main theorem applies to a number ofcovariance models found in applications. For example, well-behaved Toeplitzmatrices as well as covariance matrices whose spectral distribution is a sum ofatoms (under some conditions on the mass of the atoms) are among the models thetheorem can handle. Generalizations of the theorem to certain spiked versionsof our models and a.s. results about the largest eigenvalue are given. We alsodiscuss a simple corollary that does not require normality of the entries ofthe data matrix and some consequences for applications in multivariatestatistics.

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