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

We establish characterization results for the ergodicity of symmetric α–stable (SαS) and α–Fréchet max–stable stationary random fields. We first show that the result of Samorodnitsky [35] remains valid in the multiparameter setting, i.e., 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. The similarity of the spectral representations for sum– and max–stable random fields yields parallel characterization results in the max–stable setting. By establishing multiparameter versions of Stochastic and Birkhoff Ergodic Theorems, we give a criterion for ergodicity of these random fields which is valid for all dimensions and new even in the one–dimensional case. We also prove the equivalence of ergodicity and weak mixing for the general class of positively dependent random fields. AMS 2000 subject classifications: Primary 60G10, 60G52, 60G60; secondary 37A40, 37A50.