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

Recently Johansson (21) and Johnstone (16) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrixXgX(X X) converges to theTracy–Widom lawas n, p (the dimensions ofX) tend to . in some ratio n/pQ c > 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 nonGaussian entries, provided n−p=O(p).We also prove that lmax [ (n+p) +O(p log(p)) (a.e.) for general c > 0.