Ergodic Properties of Sum- and Max- Stable Stationary Random Fields via Null and Positive Group Actions

Details Ergodic Properties of Sum- and Max- Stable Stationary Random Fields via Null and Positive Group Actions Ergodic Properties of Sum- and Max- Stable Stationary Random Fields via Null and Positive Group Actions

2009-11-03

arxiv.org/abs/0911.0610v2

Mathematics

Probability

To appear in Annals of Probability

Scientific paper

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(2005) remains valid in the multiparameter setting, i.e., 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(1971), which is dimension-free and different from the one used by Samorodnitsky.

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