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Group-Theoretic Dimension of Stationary Symmetric α-Stable Random Fields

Author

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  • Arijit Chakrabarty

    (Indian Statistical Institute)

  • Parthanil Roy

    (Indian Statistical Institute)

Abstract

The growth rate of the partial maximum of a stationary stable process was first studied in the works of Samorodnitsky (Ann. Probab. 32:1438–1468, 2004; Adv. Appl. Probab. 36:805–823, 2004), where it was established, based on the seminal works of Rosiński (Ann. Probab. 23:1163–1187, 1995; 28:1797–1813, 2000), that the growth rate is connected to the ergodic-theoretic properties of the flow that generates the process. The results were generalized to the case of stable random fields indexed by ℤ d in Roy and Samorodnitsky (J. Theor. Probab. 21:212–233, 2008), where properties of the group of nonsingular transformations generating the stable process were studied as an attempt to understand the growth rate of the partial maximum process. This work generalizes this connection between stable random fields and group theory to the continuous parameter case, that is, to fields indexed by ℝ d .

Suggested Citation

  • Arijit Chakrabarty & Parthanil Roy, 2013. "Group-Theoretic Dimension of Stationary Symmetric α-Stable Random Fields," Journal of Theoretical Probability, Springer, vol. 26(1), pages 240-258, March.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:1:d:10.1007_s10959-011-0371-6
    DOI: 10.1007/s10959-011-0371-6
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    References listed on IDEAS

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    1. Hardin, Clyde D., 1982. "On the spectral representation of symmetric stable processes," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 385-401, September.
    2. Parthanil Roy & Gennady Samorodnitsky, 2008. "Stationary Symmetric α-Stable Discrete Parameter Random Fields," Journal of Theoretical Probability, Springer, vol. 21(1), pages 212-233, March.
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    Cited by:

    1. Parthanil Roy, 2017. "Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 513-540, December.
    2. Zaoli Chen & Gennady Samorodnitsky, 2020. "Extreme Value Theory for Long-Range-Dependent Stable Random Fields," Journal of Theoretical Probability, Springer, vol. 33(4), pages 1894-1918, December.

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