A universality result for the smallest eigenvalues of certain sample covariance matrices

TL;DR

This paper demonstrates that, under specific conditions, the smallest eigenvalue of certain sample covariance matrices converges to the Tracy--Widom distribution, extending known results about the largest eigenvalue.

Contribution

It establishes a universality result for the smallest eigenvalues of sample covariance matrices, complementing existing work on the largest eigenvalues.

Findings

Smallest eigenvalues converge to Tracy--Widom distribution

Results hold under technical assumptions and bounded aspect ratio

Complements prior work on largest eigenvalues

Abstract

After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy--Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Peche.