Tail processes and tail measures: An approach via Palm calculus

TL;DR

This paper develops an intrinsic approach to studying tail fields of stationary random fields using Palm calculus, characterizing tail measures, spectral representations, and extremal indices in a general group setting.

Contribution

It introduces a novel Palm calculus framework for tail measures of stationary random fields on general groups, extending extreme value theory tools.

Findings

Characterization of mass-stationarity via Mecke equation

Homogeneity of the tail measure established

Spectral and shift representations of tail measures derived

Abstract

Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group $\mathbb{G}$. The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure $\xi$ associated with a stationary (measurable) random field $Y=(Y_s)_{s\in \mathbb{G}}$. It is important to allow the underlying stationary measure to be $\sigma$-finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index.