On the spectral norm of large heavy-tailed random matrices with strongly dependent rows and columns

TL;DR

This paper investigates the spectral norm behavior of large heavy-tailed random matrices with dependent rows and columns, providing bounds for their covariance matrices when dimensions grow large.

Contribution

It introduces a new random matrix ensemble constructed via a 2D linear filter on iid variables with infinite fourth moments, analyzing its spectral properties.

Findings

Provides asymptotic bounds for the spectral norm of the sample covariance matrix.

Extends understanding of heavy-tailed matrices with dependent structures.

Addresses matrices with infinite fourth moments.

Abstract

We study a new random matrix ensemble $X$ which is constructed by an application of a two dimensional linear filter to a matrix of iid random variables with infinite fourth moments. Our result gives asymptotic lower and upper bounds for the spectral norm of the (centered) sample covariance matrix $XX^\T$ when the number of columns as well es the number of rows of $X$ tend to infinity.