Cluster Random Fields and Random-Shift Representations

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 Rosiński 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.

Accepted J. Th. Probability

References in corpus (6)

• Stationary max-stable fields associated to negative definite functions
• Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes
• The tail process and tail measure of continuous time regularly varying stochastic processes
• Tail Measures and Regular Variation
• On the speed of convergence of discrete Pickands constants to continuous ones
• Shift-invariant homogeneous classes of random fields