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Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions. (English) Zbl 1273.60062
Authors’ abstract: We establish characterization results for the ergodicity of stationary symmetric $$\alpha$$-stable ($$\text{S}\alpha\text{S}$$) and $$\alpha$$-Fréchet random fields. We show that the result of G. Samorodnitsky [Ann. Probab. 33, No. 5, 1781–1803 (2005; Zbl 1080.60033)] remains valid in the multiparameter setting, that is, a stationary $$\text{S}\alpha\text{S}$$ ($$0<\alpha<2$$) 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 W. Takahashi [Kōdai Math. Sem. Rep. 23, 131–143 (1971; Zbl 0219.47035)], which is dimension-free and different from the one used by Samorodnitsky.

##### MSC:
 60G60 Random fields 60G10 Stationary stochastic processes 60G52 Stable stochastic processes 37A40 Nonsingular (and infinite-measure preserving) transformations 37A50 Dynamical systems and their relations with probability theory and stochastic processes
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