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- Publisher
- Cambridge University Press
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust
- ISSN
- 1475-6072
- eISSN
- 0021-9002
- DOI
- 10.1017/jpr.2022.52
- Publisher site
- See Article on Publisher Site
Abstract
Abstract This article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster, and then we apply the developed theory to determine the limiting distribution of $\max\{X_j\,:\, j=0,\ldots, K(t)\}$ , where K(t) is the number of independent and identically distributed observations $(X_j)$ arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.
Journal
Journal of Applied Probability – Cambridge University Press
Published: Jun 1, 2023
Keywords: Renewal cluster processes; Poisson cluster processes; Hawkes process; maximal claim size; extreme value distributions; random maxima; limit theorems; 60G70; 60G55; 91G30