Universality of local eigenvalue statistics for some sample covariance matrices

We consider random, complex sample covariance matrices 1 N X * X , where X is a p × N random matrix with i.i.d. entries of distribution μ. It has been conjectured that both the distribution of the distance between nearest neighbor eigenvalues in the bulk and that of the smallest eigenvalues become, in the limit N → ∞, p N → 1, the same as that identified for a complex Gaussian distribution μ. We prove these conjectures for a certain class of probability distributions μ.