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

Anja Janssen () and Johan Segers ()
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Anja Janssen: Otto-von-Guericke University Magdeburg
Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2022020, LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

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.

Keywords: Extremal clusters; Extremal index; Inspection paradox; Point processes; Stationary time series; Time series extremes (search for similar items in EconPapers)
Pages: 13
Date: 2022-07-22
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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvad:2022020

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