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

Janßen Anja () and Segers Johan ()
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Janßen Anja: Otto-von-Guericke University Magdeburg, FMA, 39106 Magdeburg, Germany
Segers Johan: UCLouvain, LIDAM/ISBA, 1348 Louvain-la-Neuve, Belgium

Dependence Modeling, 2024, vol. 12, issue 1, 12

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.

Keywords: extremal clusters; extremal index; inspection paradox; point processes; stationary time series; time series extremes (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:12:y:2024:i:1:p:12:n:1

DOI: 10.1515/demo-2023-0109

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