Random covariance matrices: Universality of local statistics of eigenvalues up to the edge

Abstract: We study the universality of the eigenvalue statistics of the covariance matrices $\frac{1}{n}M^* M$ where $M$ is a large $p\times n$ matrix obeying condition $\bf{C1}$. In particular, as an application, we prove a variant of universality results regarding the smallest singular value of $M_{p,n}$. This paper is an extension of the results in \cite{tvcovariance} from the bulk of the spectrum up to the edge.