Invariance properties of limiting point processes and applications to clusters of extremes
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 explo...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-02-01
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| Series: | Dependence Modeling |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/demo-2023-0109 |
| _version_ | 1827353236705640448 |
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| author | Janßen Anja Segers Johan |
| author_facet | Janßen Anja Segers Johan |
| author_sort | Janßen Anja |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T03:22:07Z |
| format | Article |
| id | doaj.art-f96dd0142c894ed29a22b1fb674d990b |
| institution | Directory Open Access Journal |
| issn | 2300-2298 |
| language | English |
| last_indexed | 2024-03-08T03:22:07Z |
| publishDate | 2024-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Dependence Modeling |
| spelling | doaj.art-f96dd0142c894ed29a22b1fb674d990b2024-02-12T09:11:54ZengDe GruyterDependence Modeling2300-22982024-02-01121959710.1515/demo-2023-0109Invariance properties of limiting point processes and applications to clusters of extremesJanßen Anja0Segers Johan1Otto-von-Guericke University Magdeburg, FMA, 39106 Magdeburg, GermanyUCLouvain, LIDAM/ISBA, 1348Louvain-la-Neuve, BelgiumMotivated 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.https://doi.org/10.1515/demo-2023-0109extremal clustersextremal indexinspection paradoxpoint processesstationary time seriestime series extremes60g1060g70 |
| spellingShingle | Janßen Anja Segers Johan Invariance properties of limiting point processes and applications to clusters of extremes Dependence Modeling extremal clusters extremal index inspection paradox point processes stationary time series time series extremes 60g10 60g70 |
| title | Invariance properties of limiting point processes and applications to clusters of extremes |
| title_full | Invariance properties of limiting point processes and applications to clusters of extremes |
| title_fullStr | Invariance properties of limiting point processes and applications to clusters of extremes |
| title_full_unstemmed | Invariance properties of limiting point processes and applications to clusters of extremes |
| title_short | Invariance properties of limiting point processes and applications to clusters of extremes |
| title_sort | invariance properties of limiting point processes and applications to clusters of extremes |
| topic | extremal clusters extremal index inspection paradox point processes stationary time series time series extremes 60g10 60g70 |
| url | https://doi.org/10.1515/demo-2023-0109 |
| work_keys_str_mv | AT janßenanja invariancepropertiesoflimitingpointprocessesandapplicationstoclustersofextremes AT segersjohan invariancepropertiesoflimitingpointprocessesandapplicationstoclustersofextremes |