Cluster Random Fields and Random-Shift Representations

TL;DR

This paper introduces cluster random fields (CRFs) in an abstract framework, highlighting their importance in constructing shift-generated classes of $oldsymbol{ extit{ extalpha}}$-homogeneous random fields and exploring their relations with tail and spectral tail RFs.

Contribution

It provides a new abstract formulation of CRFs and demonstrates their role in constructing $ extalpha$-homogeneous RFs, along with applications to extremal indices and max-stable RFs.

Findings

CRFs are crucial for constructing shift-generated classes of $ extalpha$-homogeneous RFs.

New representations of extremal functional indices are derived.

Applications include purely dissipative max-stable RFs.

Abstract

Cluster random fields (CRFs) play a crucial role in the study of extremes of stationary regularly varying random fields (RFs). In particular, they appear in the Rosi\'nski representation of max-stable and $\alpha$-stable RFs. In this contribution we introduce CRFs in an abstract setting proving that they are crucial for the construction of shift-generated classes of $\alpha$-homogeneous RFs. Further, we investigate the relations between CRFs, tail RFs} and spectral tail RFs. Applications discussed in this contribution include new representations of extremal functional indices and purely dissipative max-stable RFs.