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Maxima of stable random fields, nonsingular actions and finitely generated abelian groups: A survey

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  • Parthanil Roy

    (Indian Statistical Institute)

Abstract

This is a self-contained introduction to the applications of ergodic theory of nonsingular (also known as quasi-invariant) group actions and the structure theorem for finitely generated abelian groups on the extreme values of stationary symmetric stable random fields indexed by ℤ d . It is based on a mini course given in the http://www.isid.ac.in/ lps16/PastWebsites/LPS8/index.html Eighth Lectures on Probability and Stochastic Processes (held in the Bangalore Centre of Indian Statistical Institute during December 6–10, 2013) except that a few recent references have been added in the concluding part. This article is a survey of existing work and the proofs are therefore skipped or briefly outlined.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:indpam:v:48:y:2017:i:4:d:10.1007_s13226-017-0243-6
    DOI: 10.1007/s13226-017-0243-6
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    References listed on IDEAS

    as
    1. Fasen, Vicky & Roy, Parthanil, 2016. "Stable random fields, point processes and large deviations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 832-856.
    2. Dombry, Clément & Kabluchko, Zakhar, 2017. "Ergodic decompositions of stationary max-stable processes in terms of their spectral functions," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1763-1784.
    3. Takashi Owada, 2016. "Limit Theory for the Sample Autocovariance for Heavy-Tailed Stationary Infinitely Divisible Processes Generated by Conservative Flows," Journal of Theoretical Probability, Springer, vol. 29(1), pages 63-95, March.
    4. Stoev, Stilian A., 2008. "On the ergodicity and mixing of max-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1679-1705, September.
    5. 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.
    6. 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.
    7. Hardin, Clyde D., 1982. "On the spectral representation of symmetric stable processes," Journal of Multivariate Analysis, Elsevier, vol. 12(3), pages 385-401, September.
    8. Resnick, Sidney & Samorodnitsky, Gennady, 2004. "Point processes associated with stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 191-209, December.
    9. Kabluchko, Zakhar & Schlather, Martin, 2010. "Ergodic properties of max-infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 281-295, March.
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