Invariance properties of limiting point processes and applications to clusters of extremes

TL;DR

This paper studies the invariance properties of cluster processes derived from stationary time series, introducing notions of cluster sizes and clarifying their relationship with the extremal index in extreme value theory.

Contribution

It introduces a general framework for cluster processes, explores invariance properties, and clarifies the relationship between extremal index and cluster size distributions.

Findings

Invariance properties of cluster processes are established.

Distinction between inspected and typical cluster sizes is clarified.

The extremal index is related to the expected typical cluster size, not the mean cluster size.

Abstract

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.