On extremes of random clusters and marked renewal cluster processes

TL;DR

This paper investigates the extreme values of observations in clustered processes, deriving their limiting distributions and applying the theory to models like the marked Hawkes process.

Contribution

It develops a general framework for the distribution of extremes in marked renewal cluster processes, including specific models like the Hawkes process.

Findings

Derived the limiting distribution of maxima in clustered processes.

Analyzed tail behavior of individual clusters.

Applied results to Poisson cluster models like the Hawkes process.

Abstract

The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting distribution of $\max\{X_j: j=0,\ldots, K(t)\}$, where $K(t)$ is the number of i.i.d. observations $(X_j)$ arriving up to the time $t$ according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.