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

Author

Listed:
  • Janßen Anja

    (Otto-von-Guericke University Magdeburg, FMA, 39106 Magdeburg, Germany)

  • Segers Johan

    (UCLouvain, LIDAM/ISBA, 1348 Louvain-la-Neuve, Belgium)

Abstract

Motivated by examples from extreme value theory, but without using the theory of regularly varying time series or any assumptions about the marginal distribution, 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 that are implied by stationarity of the underlying time series. Of particular interest in applications are the cluster size distributions, and we derive general properties and interconnections between the size of an inspected and a typical cluster. 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, 2024. "Invariance properties of limiting point processes and applications to clusters of extremes," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-12, January.
    • Handle: RePEc:vrs:demode:v:12:y:2024:i:1:p:12:n:1
    DOI: 10.1515/demo-2023-0109
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    References listed on IDEAS

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    1. Cirillo, Pasquale & Hüsler, Jürg, 2009. "An urn approach to generalized extreme shock models," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 969-976, April.
    2. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    3. 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.
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