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On the total claim amount for marked Poisson cluster models

Part of: Limit theorems Mathematical economics Stochastic processes

Published online by Cambridge University Press:  07 August 2019

Bojan Basrak ,
Olivier Wintenberger  and
Petra Žugec
Bojan Basrak*
Affiliation:
University of Zagreb
Olivier Wintenberger*
Affiliation:
Sorbonne Université
Petra Žugec*
Affiliation:
University of Zagreb
*
*Postal address: Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb, Croatia.
**Postal address: LPSM, Sorbonne Université, 4 Place Jussieu, F-75005, Paris, France.
***Postal address: Faculty of Organization and Informatics, University of Zagreb, Pavlinska 2, Varaždin, Croatia. Email address: [email protected]
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Abstract

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence the arrival rate of future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or, alternatively, tends in distribution to an infinite-variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes process as our key example.

Keywords

Poisson cluster processlimit theoremHawkes processtotal claim amountcentral limit theoremstable random variable

MSC classification

Primary: 91B30: Risk theory, insurance
Secondary: 60F05: Central limit and other weak theorems 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 541 - 569
Copyright
© Applied Probability Trust 2019 

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