ERGODIC PROPERTIES OF SUM- AND MAX-STABLE STATIONARY RANDOM FIELDS VIA NULL AND POSITIVE GROUP ACTIONS

作者：Wang Yizao^*; Roy Parthanil; Stoev Stilian A

来源：Annals of Probability, 2013, 41(1): 206-228.

DOI：10.1214/11-AOP732

• Stable
• max-stable
• random field
• ergodic theory
• nonsingular group action
• null action
• positive action
• ergodicity

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 [Kadai Math. Sem. Rep. 23 (1971) 131-143], which is dimension-free and different from the one used by Samorodnitsky.

• 出版日期2013-1