Spectral measure of heavy tailed band and covariance random matrices

TL;DR

This paper investigates the asymptotic spectral behavior of large symmetric matrices with heavy-tailed entries, establishing convergence of their spectral measures and analyzing the limiting spectral density for covariance matrices.

Contribution

It introduces a new framework for analyzing spectral measures of heavy-tailed random matrices with deterministic perturbations, including covariance matrices.

Findings

Spectral measures converge to a well-characterized limit.

Derived the almost sure limiting spectral density for heavy-tailed covariance matrices.

Provided a characterization of the limiting distribution for scaled and perturbed matrices.

Abstract

We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $\mu$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N by N symmetric matrix $Y_N^\sigma$ whose (i,j) entry is $\sigma(i/N,j/N)X_{ij}$ where $(X_{ij}, 0<i<j+1<\infty)$ is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an $\alpha$-stable law, $0<\alpha<2$, and $\sigma$ is a deterministic function. For a random diagonal $D_N$ independent of $Y_N^\sigma$ and with appropriate rescaling $a_N$, we prove that the distribution $\mu$ of $a_N^{-1}Y_N^\sigma + D_N$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.