Eigenvalues of sample covariance matrices of non-linear processes with infinite variance

TL;DR

This paper investigates the asymptotic behavior of the largest eigenvalues of heavy-tailed sample covariance matrices derived from non-linear, infinite variance processes, with applications to stochastic volatility and GARCH models.

Contribution

It provides a theoretical framework for understanding the eigenvalues of covariance matrices from non-linear heavy-tailed processes with infinite variance.

Findings

Asymptotic distribution of the largest eigenvalues derived

Results applicable to stochastic volatility and GARCH processes

Establishment of conditions under which eigenvalues exhibit specific behaviors

Abstract

We study the $k$-largest eigenvalues of heavy-tailed sample covariance matrices of the form $\bX\bX^\T$ in an asymptotic framework, where the dimension of the data and the sample size tend to infinity. To this end, we assume that the rows of $\bX$ are given by independent copies of some stationary process with regularly varying marginals with index $\alpha\in(0,2)$ satisfying large deviation and mixing conditions. We apply these general results to stochastic volatility and GARCH processes.