On limiting cluster size distributions for processes of exceedances for stationary sequences

TL;DR

This paper demonstrates that for any specified distribution on natural numbers, a stationary sequence can be constructed so that its exceedance process converges to a compound Poisson process with that distribution, highlighting the flexibility in cluster size distributions.

Contribution

It shows the existence of stationary sequences with prescribed cluster size distributions in their exceedance processes, extending understanding of possible limit behaviors.

Findings

Any distribution on natural numbers can be realized as the cluster size distribution of a stationary sequence.

The limit process of exceedances can be tailored to match any desired compound Poisson law.

The result broadens the scope of possible limit behaviors in extreme value theory for stationary sequences.

Abstract

It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate that, for any given distribution G on the natural numbers, there exists a stationary sequence for which the compounding law of this limiting process of exceedances will coincide with G.