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On extremes of random clusters and marked renewal cluster processes

Part of: Stochastic processes Mathematical finance

Published online by Cambridge University Press:  09 December 2022

Bojan Basrak ,
Nikolina Milinčević  and
Petra Žugec
Bojan Basrak*
Affiliation:
University of Zagreb
Nikolina Milinčević*
Affiliation:
University of Zagreb
Petra Žugec*
Affiliation:
University of Zagreb
*
*Postal address: Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb, Croatia.
*Postal address: Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb, Croatia.
****Postal address: Faculty of Organization and Informatics, University of Zagreb, Pavlinska 2, Varaždin, Croatia. Email address: [email protected]
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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.

Keywords

Renewal cluster processesPoisson cluster processesHawkes processmaximal claim sizeextreme value distributionsrandom maximalimit theorems

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G55: Point processes 91G30: Interest rates (stochastic models)
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 367 - 381
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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