Some variations on the extremal index

TL;DR

This paper revisits the extremal index for stationary sequences, especially heavy-tailed time series, providing explicit representations and applications to AR(1) processes and stochastic recurrences.

Contribution

It offers new explicit formulas for the extremal index in regularly varying sequences and explores its diverse representations and applications.

Findings

Explicit expressions for the extremal index in regularly varying sequences

Representation of the limiting cluster structure of extremes

Application to AR(1) and stochastic recurrence models

Abstract

We re-consider Leadbetter's extremal index for stationary sequences. It has interpretation as reciprocal of the expected size of an extremal cluster above high thresholds. We focus on heavy-tailed time series, in particular on regularly varying stationary sequences, and discuss recent research in extreme value theory for these models. A regularly varying time series has multivariate regularly varying finite-dimensional distributions. Thanks to results by Basrak and Segers we have explicit representations of the limiting cluster structure of extremes, leading to explicit expressions of the limiting point process of exceedances and the extremal index as a summary measure of extremal clustering. The extremal index appears in various situations which do not seem to be directly related, like the convergence of maxima and point processes. We consider different representations of the extremal index which arise from the considered context. We discuss the theory and apply it to a regularly varying AR(1) process and the solution to an affine stochastic recurrence equation