  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.
