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 ($\mathrm{S}\alpha\mathrm{S}$) and $\alpha$-Fréchet 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 $\mathrm{S}\alpha\mathrm{S}$ ($0<\alpha<2$) 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 [Kōdai Math. Sem. Rep. 23 (1971) 131–143], which is dimension-free and different from the one used by Samorodnitsky.