Long Range Dependence for Stable Random Processes

TL;DR

This paper explores long and short memory properties in heavy-tailed stable and max-stable processes using covariance of excursions and extremal coefficients, providing conditions for dependence based on kernel integrability.

Contribution

It introduces new criteria for long and short range dependence in $eta$-stable and max-stable processes, extending dependence analysis to heavy-tailed models.

Findings

Sufficient conditions for dependence in $eta$-stable moving averages.

Necessary and sufficient conditions for max-stable processes.

Dependence characterized via kernel integrability and extremal coefficients.

Abstract

We investigate long and short memory in $\alpha$-stable moving averages and max-stable processes with $\alpha$-Fr\'echet marginal distributions. As these processes are heavy-tailed, we rely on the notion of long range dependence suggested by Kulik and Spodarev (2019) based on the covariance of excursions. Sufficient conditions for the long and short range dependence of $\alpha$-stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max-stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence.