Random covariance matrices: Universality of local statistics of eigenvalues

TL;DR

This paper proves a universality result for the local eigenvalue statistics of large covariance matrices with i.i.d. entries, extending previous results to broader moment conditions.

Contribution

It establishes a Four Moment theorem for covariance matrices, enabling universality of local eigenvalue statistics under mild assumptions.

Findings

Universality of local eigenvalue statistics for covariance matrices.

Extension of previous results to finite moment conditions.

Development of a Four Moment theorem analogous to Wigner matrices.

Abstract

We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a large 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, 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.