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Ergodic properties of max-infinitely divisible processes

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  • Kabluchko, Zakhar
  • Schlather, Martin

Abstract

We prove that a stationary max-infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesàro summable to 0). These criteria are applied to some classes of max-infinitely divisible processes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:3:p:281-295
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    References listed on IDEAS

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    1. Cambanis, Stamatis & Hardin, Clyde D. & Weron, Aleksander, 1987. "Ergodic properties of stationary stable processes," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 1-18, February.
    2. Gross, Aaron & Robertson, James B., 1993. "Ergodic properties of random measures on stationary sequences of sets," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 249-265, June.
    3. Rosinski, Jan & Zak, Tomasz, 1996. "Simple conditions for mixing of infinitely divisible processes," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 277-288, February.
    4. Martin Schlather, 2003. "A dependence measure for multivariate and spatial extreme values: Properties and inference," Biometrika, Biometrika Trust, vol. 90(1), pages 139-156, March.
    5. Gross, Aaron, 1994. "Some mixing conditions for stationary symmetric stable stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 51(2), pages 277-295, July.
    6. 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.
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    Cited by:

    1. 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.
    2. Dombry, Clément & Eyi-Minko, Frédéric, 2012. "Strong mixing properties of max-infinitely divisible random fields," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3790-3811.
    3. Wang, Yizao, 2018. "Extremes of q-Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2979-3005.
    4. Koch, Erwan & Dombry, Clément & Robert, Christian Y., 2019. "A central limit theorem for functions of stationary max-stable random fields on Rd," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3406-3430.
    5. Raphaël Huser & Marc G. Genton, 2016. "Non-Stationary Dependence Structures for Spatial Extremes," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 470-491, September.
    6. Erwan Koch, 2018. "Extremal dependence and spatial risk measures for insured losses due to extreme winds," Papers 1804.05694, arXiv.org, revised Dec 2019.
    7. 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.
    8. Padoan, Simone A., 2013. "Extreme dependence models based on event magnitude," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 1-19.

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