The necessary and sufficient conditions in the Marchenko-Pastur theorem

TL;DR

This paper establishes that a weak concentration property of quadratic forms is both necessary and sufficient for the Marchenko-Pastur theorem to hold for certain sample covariance matrices, providing new conditions for its validity.

Contribution

It proves the equivalence between weak concentration of quadratic forms and the Marchenko-Pastur theorem's applicability, offering general conditions for the property.

Findings

Weak concentration property is necessary and sufficient for the theorem.

General conditions are provided to guarantee the weak concentration.

The results extend the understanding of sample covariance matrices.

Abstract

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