Local Marchenko-Pastur law at the hard edge of the Sample Covariance ensemble

TL;DR

This paper proves that the eigenvalue density of sample covariance matrices converges to the Marchenko-Pastur law at a fine scale, establishing eigenvalue rigidity near edges under minimal moment assumptions.

Contribution

It extends local Marchenko-Pastur law results to covariance matrices with minimal moment conditions, simplifying previous methods.

Findings

Eigenvalue density converges to Marchenko-Pastur law at optimal scale.

Eigenvalue rigidity established near edges and in the bulk.

Results hold under finite 4th moment and truncation at N^{1/4}.

Abstract

Consider an $N$ by $N$ matrix $X$ of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of $X^*X$ converges to the Marchenko-Pastur law on the optimal scale with probability $1$. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under the assumption that the entries have a finite 4th moment and are truncated at $N^{1/4}$. In this work we simplify and adapt the methods from prior papers of G\"otze-Tikhomirov and Cacciapuoti-Maltsev-Schlein to covariance matrices.