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Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure

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  • Rønn-Nielsen, Anders
  • Stehr, Mads

Abstract

We consider an infinitely divisible random field indexed by Rd, d∈N, given as an integral of a kernel function with respect to a Lévy basis with a Lévy measure having a regularly varying right tail. First we show that the tail of its supremum over any bounded set is asymptotically equivalent to the right tail of the Lévy measure times the integral of the kernel. Secondly, when observing the field over an appropriately increasing sequence of continuous index sets, we obtain an extreme value theorem stating that the running supremum converges in distribution to the Fréchet distribution.

Suggested Citation

  • Rønn-Nielsen, Anders & Stehr, Mads, 2022. "Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 19-49.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:19-49
    DOI: 10.1016/j.spa.2022.04.007
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    References listed on IDEAS

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    1. Molchanov, Ilya & Strokorb, Kirstin, 2016. "Max-stable random sup-measures with comonotonic tail dependence," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2835-2859.
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    5. Kristjana Ýr Jónsdóttir & Anders Rønn-Nielsen & Kim Mouridsen & Eva B. Vedel Jensen, 2013. "Lévy-based Modelling in Brain Imaging," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 511-529, September.
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