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Nonsingular group actions and stationary S$ \alpha$S random fields

Author: Parthanil Roy
Journal: Proc. Amer. Math. Soc. 138 (2010), 2195-2202
MSC (2010): Primary 60G60; Secondary 60G70, 60G52, 37A40
Published electronically: February 2, 2010
MathSciNet review: 2596059
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Abstract: This paper deals with measurable stationary symmetric stable random fields indexed by $ \mathbb{R}^d$ and their relationship with the ergodic theory of nonsingular $ \mathbb{R}^d$-actions. Based on the phenomenal work of Rosiński (2000), we establish extensions of some structure results of stationary $ S\alpha S$ processes to $ S\alpha S$ fields. Depending on the ergodic theoretical nature of the underlying action, we observe different behaviors of the extremes of the field.

References [Enhancements On Off] (What's this?)

  • [1] J. AARONSON (1997): An Introduction to Infinite Ergodic Theory, volume 50 of Mathematical Surveys and Monographs.
    American Mathematical Society, Providence, RI. MR 1450400 (99d:28025)
  • [2] S. BANACH (1932): Théorie des óperations linéaires.
    Chelsea Publishing Co., New York. MR 0071726 (17:175h)
  • [3] D. COHN (1972): Measurable choice of limit points and the existence of separable and measurable processes.
    Z. Wahr. verw. Geb. 22:161-165. MR 0305444 (46:4574)
  • [4] S. KOLODYŃSKI AND J. ROSIŃSKI (2003): Group self-similar stable processes in $ \mathbb{R}\sp d$.
    J. Theoret. Probab. 16:855-876. MR 2033189 (2005b:60104)
  • [5] U. KRENGEL (1969): Darstellungssätze für Strömungen und Halbströmungen. II.
    Mathematische Annalen 182:1-39. MR 0296254 (45:5315)
  • [6] U. KRENGEL (1985): Ergodic Theorems.
    De Gruyter, Berlin-New York. MR 797411 (87i:28001)
  • [7] J. ROSIŃSKI (1995): On the structure of stationary stable processes.
    Annals of Probability 23:1163-1187. MR 1349166 (96k:60091)
  • [8] J. ROSIŃSKI (2000): Decomposition of stationary $ \alpha$-stable random fields.
    Annals of Probability 28:1797-1813. MR 1813849 (2002h:60070)
  • [9] E. ROY (2007): Ergodic properties of Poissonian ID processes.
    Annals of Probability 35:551-576. MR 2308588 (2008d:60054)
  • [10] P. ROY (2007): Ergodic theory, abelian groups, and point processes induced by stable random fields.
    Preprint, available at
    To appear in Annals of Probability.
  • [11] P. ROY AND G. SAMORODNITSKY (2008): Stationary symmetric $ \alpha$-stable discrete parameter random fields.
    Journal of Theoretical Probability 21:212-233. MR 2384479 (2008m:60091)
  • [12] G. SAMORODNITSKY (2004): Extreme value theory, ergodic theory, and the boundary between short memory and long memory for stationary stable processes.
    Annals of Probability 32:1438-1468. MR 2060304 (2005c:60038)
  • [13] G. SAMORODNITSKY (2004): Maxima of continuous time stationary stable processes.
    Advances in Applied Probability 36:805-823. MR 2079915 (2005j:60098)
  • [14] G. SAMORODNITSKY (2005): Null flows, positive flows and the structure of stationary symmetric stable processes.
    Annals of Probability 33:1782-1803. MR 2165579 (2006h:60066)
  • [15] G. SAMORODNITSKY AND M. TAQQU (1994): Stable Non-Gaussian Random Processes.
    Chapman and Hall, New York. MR 1280932 (95f:60024)
  • [16] D. SURGAILIS, J. ROSIŃSKI, V. MANDREKAR AND S. CAMBANIS (1993): Stable mixed moving averages.
    Probab. Theory Related Fields 97:543-558. MR 1246979 (94k:60080)
  • [17] D. SURGAILIS, J. ROSIŃSKI, V. MANDREKAR AND S. CAMBANIS (1998): On the mixing structure of stationary increment and self-similar $ S\alpha S$ processes.

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Additional Information

Parthanil Roy
Affiliation: Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824-1027

Keywords: Random field, stable process, extreme value theory, maxima, ergodic theory, nonsingular group action, dissipative, conservative
Received by editor(s): December 30, 2008
Received by editor(s) in revised form: October 9, 2009
Published electronically: February 2, 2010
Additional Notes: The author was supported in part by NSF grant DMS-0303493 and NSF training grant “Graduate and Postdoctoral Training in Probability and Its Applications” at Cornell University, the RiskLab of the Department of Mathematics, ETH Zurich, and a start-up grant from Michigan State University.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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