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Large deviations of heavy-tailed random sums with applications in insurance and finance

Part of: Applications Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

C. Klüppelberg  and
T. Mikosch
C. Klüppelberg*
Affiliation:
Johannes Gutenberg University Mainz
T. Mikosch*
Affiliation:
University of Groningen
*
∗ Postal address: Department of Mathematics, Johannes Gutenberg University Mainz, D-55099 Mainz, Germany.
∗∗ Postal address: Department of Mathematics, University of Groningen, P.O. Box 800, NL-9700 Groningen, The Netherlands.
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Abstract

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Keywords

COMPOUND POISSON PROCESSEXTREME VALUE THEORYFINANCIAL RISKFUTURESHIGH DENSITY DATAINSURANCE RISKLARGE DEVIATIONSREGULAR VARIATIONREINSURANCERENEWAL COUNTING PROCESSSUBEXPONENTIAL DISTRIBUTIONS

MSC classification

Primary: 60F10: Large deviations 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Secondary: 60K10: Applications (reliability, demand theory, etc.) 62P05: Applications to actuarial sciences and financial mathematics

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 293 - 308
Copyright
Copyright © Applied Probability Trust 1997 

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