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Cluster Random Fields and Random-Shift Representations

Enkelejd Hashorva

TL;DR

This work develops an abstract theory of cluster random fields ($CRFs$) and their role in shift-generated $α$-homogeneous classes of random fields. It establishes precise connections between $CRFs$, tail and spectral tail RFs, and the Rosiński representation of max-stable and $α$-stable RFs, enabling new random-shift representations and diverse constructions of $CRFs$. The paper provides explicit CRF constructions for purely dissipative classes, derives representations for extremal indices, and applies these tools to Brown-Resnick models and $m$-approximation schemes, thereby enriching the toolkit for extremal analysis of stationary regularly varying RFs. The results have potential impact on extremal index estimation, functional index representations, and tractable simulations for high-dimensional extreme value modelling via flexible CRF-based 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 $α$-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 $α$-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.

Cluster Random Fields and Random-Shift Representations

TL;DR

This work develops an abstract theory of cluster random fields () and their role in shift-generated -homogeneous classes of random fields. It establishes precise connections between , tail and spectral tail RFs, and the Rosiński representation of max-stable and -stable RFs, enabling new random-shift representations and diverse constructions of . The paper provides explicit CRF constructions for purely dissipative classes, derives representations for extremal indices, and applies these tools to Brown-Resnick models and -approximation schemes, thereby enriching the toolkit for extremal analysis of stationary regularly varying RFs. The results have potential impact on extremal index estimation, functional index representations, and tractable simulations for high-dimensional extreme value modelling via flexible CRF-based 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 -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 -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.
Paper Structure (15 sections, 16 theorems, 137 equations)

This paper contains 15 sections, 16 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous, anchoring and involution maps
  4. Local, spectral tail and tail RFs
  5. Characterisation of pure conservativity/dissipativity
  6. Main Results
  7. Shift-generated $\alpha$-homogeneous classes and CRFs
  8. Constructions of CRFs of purely dissipative $\mathcal{C}_{\kappa} [Z ]$'s
  9. Applications
  10. $\mathcal{L}$-extremal index
  11. Max-stable RFs
  12. Shift-generated Brown-Resnick $\mathcal{C}_{\kappa} [Z ]$'s
  13. $m$-approximation
  14. Proofs
  15. Technical Results

Key Result

Lemma 1.7

Let $\mathcal{C}_{\kappa} [Z ]$ be shift invariant. It is purely conservative/dissipative if and only if (iff) the max-stable RF $X_\kappa$ with representer $Z^{1/\alpha}_\kappa$ and de Haan representation as in eq1 is purely conservative/dissipative.

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Example 1.5
  • Definition 1.6
  • Lemma 1.7
  • Lemma 1.8
  • Example 1.9
  • Example 1.10
  • ...and 34 more