| Samorodnitsky (2005) obtained a decomposition of stationary symmetric stable processes and characterized the ergodicity of such processes using this decomposition. We extend this work to stationary symmetric stable random fields based on the work of Rosinski (2000) relating the structure of stable random fields to nonsingular group actions. The main hindrance of this extension is the unavailability of the ergodic theory results of Krengel (1967) for group actions. We prove a multiparameter ergodic theorem and use some results of Takahashi (1971) to establish this extension. Similar results are also obtained for stationary max-stable random fields. (This talk is based on an ongoing joint work with Yizao Wang and Stilian A. Stoev.)
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