A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices

Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X*X(X ^t X) converges to the Tracy–Widom law as n,p (the dimensions of X) tend to in some ratio n/p>0. We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n–p=O(p ^1/3). We also prove that _max(n ^1/2+p ^1/2)²+O(p ^1/2 log(p)) (a.e.) for general >0.