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

TL;DR

This paper extends the universality results for the distribution of the largest eigenvalues of Wishart matrices, showing convergence to Tracy-Widom law for multiple eigenvalues and non-Gaussian entries, with bounds on the maximum eigenvalue.

Contribution

It proves joint convergence of multiple eigenvalues to Tracy-Widom distribution and extends universality to non-Gaussian Wishart matrices under certain conditions.

Findings

Joint distribution of top eigenvalues converges to Tracy-Widom law.

Universality extends to non-Gaussian entries with $ n-p=O(p^{1/3}) $.

Maximum eigenvalue is bounded by $(n^{1/2}+p^{1/2})^2 + O(p^{1/2}\log p)$ almost surely.

Abstract

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 $ \infty $ in some ratio $ n/p \to \gamma>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(p^{1/3}) . $ We also prove that $ \lambda_{max} \leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\*\log(p)) $ (a.e.) for general $ \gamma >0.$