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

Yizao Wang, Parthanil Roy, Stilian Stoev

• We establish characterization results for the ergodicity of stationary symmetric α-stable (SαS) and α-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 SαS (0 < α < 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.