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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.

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  • 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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    1. Fred Espen Benth & Anca Pircalabu, 2018. "A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures," Applied Mathematical Finance, Taylor & Francis Journals, vol. 25(1), pages 36-65, January.
    2. Zaoli Chen & Gennady Samorodnitsky, 2020. "Extreme Value Theory for Long-Range-Dependent Stable Random Fields," Journal of Theoretical Probability, Springer, vol. 33(4), pages 1894-1918, December.
    3. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    4. 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.
    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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    Cited by:

    1. Danijel Grahovac & Péter Kevei, 2025. "Tail Behavior and Almost Sure Growth Rate of Superpositions of Ornstein–Uhlenbeck-type Processes," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-15, March.

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