Central limit theorems for linear statistics of heavy tailed random matrices

TL;DR

This paper establishes central limit theorems for linear statistics of heavy-tailed random matrices, including those with stable law domains and exploding moments, revealing Gaussian limits with classical normalization.

Contribution

It extends CLT results to heavy-tailed matrices and adjacency matrices of Erdös-Rényi graphs, with novel normalization and eigenvalue moment CLTs.

Findings

Gaussian limit laws for heavy-tailed matrices

CLT with classical normalization for eigenvalue moments

Applicable to matrices with stable law domains and exploding moments

Abstract

We show central limit theorems (CLT) for the Stieltjes transforms or more general analytic functions of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of $\alpha$-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erd\"os-R\'enyi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues distribution. The limit laws are Gaussian, but unlike to the case of standard Wigner matrices, the normalization is the one of the classical CLT for independent random variables.