Extreme Value Theory for Spatial Random Fields – With Application to a Lévy-Driven Field

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Abstract

First, we consider a stationary random field indexed by an increasing sequence of subsets of Zd. Under certain mixing and anti–clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution. Secondly, we consider a continuous, infinitely divisible random field indexed by Rd given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light–tailed part of the field.
Original languageEnglish
JournalExtremes
Number of pages43
ISSN1386-1999
DOIs
Publication statusPublished - 7 May 2021

Bibliographical note

Epub ahead of print. Published online: 7. May 2021

Keywords

  • Extreme value theory
  • Spatial models
  • Lévy-based modeling
  • Geometric probability
  • Intrinsic volumes
  • Convolution equivalence
  • Random fields

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