Random covariance matrices: Universality of local statistics of eigenvalues

We study the eigenvalues of the covariance matrix $\frac{1}{n}{M}^{\ast }M$ of a large rectangular matrix $M=M_{n,p}=({\zeta }_{ij}{)}_{1\le i\le p;1\le j\le n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite $C_0$th moment for some sufficiently large constant $C_0$. 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 $C_0$th moment rather than exponential decay.