We study the eigenvalues of the covariance matrix 1/n M∗M of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and having finite C0th moment for some sufficiently large constant C0.
The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549–572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions.
As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C0th moment rather than exponential decay.
"Random covariance matrices: Universality of local statistics of eigenvalues." Ann. Probab. 40 (3) 1285 - 1315, May 2012. https://doi.org/10.1214/11-AOP648