Extreme eigenvalue statistics of $m$-dependent heavy-tailed matrices

TL;DR

This paper studies the behavior of the largest eigenvalues in m-dependent heavy-tailed random matrices, extending previous independent-entry results and showing they form a Poisson cluster process.

Contribution

It introduces analysis of m-dependent heavy-tailed matrices, broadening understanding beyond independent-entry models and establishing the Poisson cluster process limit.

Findings

Largest eigenvalues follow a Poisson cluster process

Extension of heavy-tailed matrix results to m-dependent entries

Provides new insights into eigenvalue extremal behavior

Abstract

We analyze the largest eigenvalue statistics of m-dependent heavy-tailed Wigner matrices as well as the associated sample covariance matrices having entry-wise regularly varying tail distributions with parameter $0<\alpha<4$. Our analysis extends results in the previous literature for the corresponding random matrices with independent entries above the diagonal, by allowing for m-dependence between the entries of a given matrix. We prove that the limiting point process of extreme eigenvalues is a Poisson cluster process.