# Extremal Index for a Class of Heavy-Tailed Stochastic Processes in Risk Theory

## Abstract

Extreme values for dependent data corresponding to high threshold exceedances may occur in clusters, i.e., in groups of observations of different sizes. In the context of stationary sequences, the so-called extremal index measures the strength of the dependence and may be useful to estimate the average length of such clusters. This is of particular interest in risk theory where public institutions would like to predict the replications of rare events, in other words, to understand the dependence structure of extreme values. In this contribution, we characterize the extremal index for a class of stochastic processes that naturally appear in risk theory under the assumption of heavy-tailed jumps. We focus on Shot Noise type-processes and we weaken the usual assumptions required on the Shot functions. Precisely, they may be possibly random with not necessarily compact support and we do not make any assumption regarding the monotonicity. We bring to the fore the applicability of the result on a Kinetic Dietary Exposure Model used in modeling pharmacokinetics of contaminants.

## Notes

### Acknowledgements

Charles Tillier would like to thank Patrice Bertail and Philippe Soulier, both Professors at Paris Nanterre University for insightful comments and discussions which led to an improvement of this work. Financial supports by the ANR network AMERISKA ANR 14 CE20 0006 01 and the Labex MME-DII are also gratefully acknowledged by the author.

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