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Invariance properties of limiting point processes and applications to clusters of extremes

Author

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  • Janssen, Anja

    (Otto-von-Guericke University Magdeburg)

  • Segers, Johan

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

Abstract

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a “mean cluster size”, we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

Suggested Citation

  • Janssen, Anja & Segers, Johan, 2022. "Invariance properties of limiting point processes and applications to clusters of extremes," LIDAM Discussion Papers ISBA 2022020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2022020
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    References listed on IDEAS

    as
    1. Borovkov, Konstantin & Novak, Serguei, 2010. "On limiting cluster size distributions for processes of exceedances for stationary sequences," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1814-1818, December.
    2. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
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