Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices

TL;DR

This paper investigates how the largest eigenvalues of heavy-tailed random matrices behave, showing they are dominated by the largest entries when the fourth moment is absent, extending previous results.

Contribution

It extends Soshnikov's work by proving Poisson convergence of the largest eigenvalues for heavy-tailed matrices without the fourth moment.

Findings

Largest eigenvalues follow the distribution of the largest matrix entries

Poisson convergence established for heavy-tailed matrices

Results apply to real symmetric and covariance matrices

Abstract

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.