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Shift-invariant homogeneous classes of random fields

Enkelejd Hashorva

Abstract

Given an $R^d$-valued random field (rf) $Z(t),t\in T$ and an $α$-homogeneous mapping $κ$ we define the corresponding equivalent class of rf's (denoted by $K_α$) which include representers of the same tail measure $ν_Z$. When $T$ is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of $Z$. This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant $K_α$'s. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps $U$ acting on elements of $K_α$ turns out to be crucial for properties of functionals of $Z$. Applications of our findings concern max-stable and symmetric $α$-stable rf's, their maximal indices as well as their random shift-representations.

Shift-invariant homogeneous classes of random fields

Abstract

Given an -valued random field (rf) and an -homogeneous mapping we define the corresponding equivalent class of rf's (denoted by ) which include representers of the same tail measure . When is an additive group, tractable equivalent classes of interest are the shift-invariant ones, which contain in particular all independent random shifts of . This contribution is mainly concerned with the investigation of the probabilistic properties of shift-invariant 's. Important objects introduced in our setting are tail and spectral tail rf's. Further, the class of universal maps acting on elements of turns out to be crucial for properties of functionals of . Applications of our findings concern max-stable and symmetric -stable rf's, their maximal indices as well as their random shift-representations.

Paper Structure

This paper contains 14 sections, 21 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. Some classes of functionals
  5. Anchoring & involution maps
  6. Extensions of \ref{['boll']}
  7. Characterisation of shift-invariance
  8. Tail and spectral tail RF's
  9. New $\alpha$-homogeneous classes & Universal Maps
  10. Discussions
  11. Max-stable rf's
  12. Maximal indices
  13. Proofs
  14. Appendix

Key Result

Lemma 3.1

Fix $Z \in \mathfrak{W}_{\kappa}, \kappa \in \mathcal{H}_\alpha$ and suppose that ZY holds for both $\kappa$ and $\kappa_\infty$. If further then boll is equivalent with for all $(t_1 , \ldots, t_n)\in {\mathbb{T}_0}^n, n\in \mathbb{N},x\in (0,\infty)^d, \widetilde{ Z}\in \mathscr{C}_{\kappa} [Z].$

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4: Brown-Resnick $\mathscr{C}_{\kappa} [Z]$
  • Example 1.5: Brown-Resnick-Lévy $\mathscr{C}_{\kappa} [Z]$
  • Definition 1.6
  • Definition 1.7
  • Definition 2.1
  • Lemma 3.1
  • Remark 3.2
  • ...and 42 more