Random covariance matrices: Universality of local statistics of eigenvalues

Details Random covariance matrices: Universality of local statistics of eigenvalues Random covariance matrices: Universality of local statistics of eigenvalues

2009-12-07

arxiv.org/abs/0912.0966v4

Mathematics

Spectral Theory

26 pages, no figures, submitted to Annals of Probability. Some corrections and changes to the exposition

Scientific paper

We study the eigenvalues values of the covariance matrix $\frac{1}{n} M^\ast M$ of a large rectangular matrix $M = M_{n,p} = (\zeta_{ij})_{1 \leq i \leq p; 1 \leq j \leq n}$ whose entries are iid random variables of mean zero, variance one, and having finite $C_0^{th}$ moment for some sufficiently large $C_0$. The main result of this paper is a Four Moment Theorem for iid covariance matrices (analogous to the Four Moment Theorem for Wigner matrices established by the authors). Indeed, our arguments here draw heavily from those in our previous work. As in that paper, 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 Wigner matrices to the case in which the entries have finite $C_0^{th}$ moment rather than exponential decay.

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