The necessary and sufficient conditions in the Marchenko-Pastur theorem

Abstract

We show that a weak concentration property for quadratic forms of isotropic random vectors $\mathbf{x}$ is necessary and sufficient for the validity of the Marchenko-Pastur theorem for sample covariance matrices of random vectors having the form $C\mathbf{x}$, where $C$ is any rectangular matrix with orthonormal rows. We also obtain some general conditions guaranteeing the weak concentration property.