Cluster Random Fields and Random-Shift Representations

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.