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January 2013 Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions
Yizao Wang, Parthanil Roy, Stilian A. Stoev
Ann. Probab. 41(1): 206-228 (January 2013). DOI: 10.1214/11-AOP732

Abstract

We establish characterization results for the ergodicity of stationary symmetric $\alpha$-stable ($\mathrm{S}\alpha\mathrm{S}$) and $\alpha$-Fréchet 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 $\mathrm{S}\alpha\mathrm{S}$ ($0<\alpha<2$) 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 [Kōdai Math. Sem. Rep. 23 (1971) 131–143], which is dimension-free and different from the one used by Samorodnitsky.

Citation

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Yizao Wang. Parthanil Roy. Stilian A. Stoev. "Ergodic properties of sum- and max-stable stationary random fields via null and positive group actions." Ann. Probab. 41 (1) 206 - 228, January 2013. https://doi.org/10.1214/11-AOP732

Information

Published: January 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1273.60062
MathSciNet: MR3059197
Digital Object Identifier: 10.1214/11-AOP732

Subjects:
Primary: 60G10, 60G52, 60G60
Secondary: 37A40, 37A50

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 1 • January 2013
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