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We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of alarge rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$whose entries are i.i.d. random variables of mean zero, variance one, andhaving finite $C_0$th moment for some sufficiently large constant $C_0$. Themain result of this paper is a Four Moment theorem for i.i.d. covariancematrices (analogous to the Four Moment theorem for Wigner matrices establishedby the authors in [Acta Math. (2011) Random matrices: Universality of localeigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--572])). Wecan use this theorem together with existing results to establish universalityof local statistics of eigenvalues under mild conditions. As a byproduct of ourarguments, we also extend our previous results on random Hermitian matrices tothe case in which the entries have finite $C_0$th moment rather thanexponential decay.

Random covariance matrices: Universality of local statistics of eigenvalues