Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions

TL;DR

This paper characterizes the ergodic properties of sum- and max-stable stationary random fields using null and positive group actions, extending existing results to multiparameter settings and different types of stable fields.

Contribution

It extends ergodicity characterizations to multiparameter stable and max-stable fields using a new approach based on Takahashi's recurrence classification.

Findings

Ergodicity of SαS fields linked to null group actions.

Results valid for multiparameter and max-stable fields.

New dimension-free characterization of recurrence.

Abstract

We establish characterization results for the ergodicity of stationary symmetric $\alpha$-stable (S$\alpha$S) and $\alpha$-Frechet random fields. We show that the result of Samorodnitsky [Ann. Probab. 33 (2005) 1782-1803] remains valid in the multiparameter setting, that is, a stationary S$\alpha$S ($0<\alpha<2$) random field is ergodic (or, equivalently, weakly mixing) if and only if it is generated by a null group action. Similar results are also established for max-stable random fields. The key ingredient is the adaption of a characterization of positive/null recurrence of group actions by Takahashi [Kodai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free and different from the one used by Samorodnitsky.