A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices

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Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X* X (Xᵗ X) converges to the Tracy-Widom law as n, p (the dimensions of X) tend to ∞ in some ratio n/p → γ>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 matrices allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n-p =O(p¹/³). We also prove that λmax ≤ (n¹/²+p¹/²)² +O(p¹/²(p)) (a.e.) for general γ>0.