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

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.

Published in at http://dx.doi.org/10.1214/11-AOP732 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
• Ergodic properties of Poissonian ID processes
• Random rewards, fractional Brownian local times and stable self-similar processes

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