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

Yizao Wang, Parthanil Roy, Stilian A. Stoev
2013 Annals of Probability  
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
more » ... 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.
doi:10.1214/11-aop732 fatcat:rmcxmcqtyjeqrkzqonvo4wrks4