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.

Appendix

For reader’s convenience, we recall in this part important notions involved in the proof of Theorem 1. We start by the definition of the so-called “tail process” introduced recently by Basrak and Segers [3].

Definition 1 (The Tail Process)

Let $(Z_i)_{i \in \mathbb {Z}}$ be a stationary process in $\mathbb {R}^+$ and let α ∈ (0, ∞). If $(Z_i)_{i \in \mathbb {Z}}$ is jointly regularly varying with index − α, that is, all vectors of the form $(X_k,\ldots ,X_l), k \leq l \in \mathbb {Z}$ are multivariate regularly varying, then there exists a process $(Y_i)_{i \in \mathbb {Z}}$ in $\mathbb {R}^+$, called the tail process such that $\mathbb {P}(Y_0>y)=y^{- \alpha }$, y ≥ 1 and for all $(n,m)\in \mathbb {Z}^2$, n ≥ m

$$\displaystyle \begin{aligned} \lim_{z \to \infty} \mathbb{P}((z^{-1}Z_{n},\cdots,z^{-1}Z_{m}) \in \cdot \mid Z_0>z)=\mathbb{P}((Y_n,\cdots,Y_m)\in \cdot).\end{aligned}$$

We recall now the strong mixing and anti-clustering conditions.

Definition 2 (Strong Mixing Condition)

A stationary sequence $(Z_k)_{k \in \mathbb {Z}}$ is said to be strongly mixing with rate function α _h if

$$\displaystyle \begin{aligned} \sup |\mathbb{P}(A \cap B)-\mathbb{P}(A)\mathbb{P}(B)|=\alpha_h \to 0, \quad h\to \infty, \end{aligned} $$

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where the supremum is taken over all sets A ∈ σ(⋯ , Z ₋₁, Z ₀) and B ∈ σ(Z _h, Z _h+1, ⋯ )

Definition 3 (Anti-clustering Condition)

A positive stationary sequence $(Z_k)_{k \in \mathbb {Z}}$ is said to satisfy the anti-clustering condition if for all u ∈ (0, ∞),

$$\displaystyle \begin{aligned} \lim_{k \to \infty} \limsup_{n \to \infty} \mathbb{P}\left( \max_{k\leq |i| \leq r_n}Z_i>a_nu \mid Z_0>a_n u \right)=0. \end{aligned} $$

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“with (a _n) a sequence such that $\lim _{n \rightarrow \infty } n \mathbb {P}(|Z_0|>a_n) =1$” and r _n →∞ is an integer sequence such that r _n = o(n).