Extremal index for a class of heavy-tailed stochastic processes in risk theory

Extreme values for dependent data corresponding to high threshold ex-ceedences 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 paper, we characterise 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 the Kinetic Dietary Exposure Model introduced in [6] used in modeling pharmacokinetics of contaminants