Spectral analysis of spatial-sign covariance matrices for heavy-tailed data with dependence

TL;DR

This paper studies the spectral behavior of spatial-sign covariance matrices derived from heavy-tailed, dependent data, establishing limiting distributions and CLTs under near-optimal conditions.

Contribution

It provides the first analysis of spectral properties of spatial-sign covariance matrices for heavy-tailed, dependent data, including limiting spectral distribution and CLT results.

Findings

Marčenko-Pastur law holds for $oldsymbol{ ext{alpha}} ext{ } ext{geq} ext{ } 2$

CLT for linear spectral statistics valid for $oldsymbol{ ext{alpha}} ext{ } ext{greater than} ext{ } 4$

Conditions are nearly the weakest possible for such heavy-tailed data with dependence.

Abstract

This paper investigates the spectral properties of spatial-sign covariance matrices, a self-normalized version of sample covariance matrices, for data from $\alpha$-regularly varying populations with general covariance structures. By exploiting the elegant properties of self-normalized random variables, we establish the limiting spectral distribution and a central limit theorem for linear spectral statistics. We demonstrate that the Mar{\u{c}}enko-Pastur equation holds under the condition $\alpha \geq 2$, while the central limit theorem for linear spectral statistics is valid for $\alpha>4$, which are shown to be nearly the weakest possible conditions for spatial-sign covariance matrices from heavy-tailed data in the presence of dependence.