Extremes of L\'evy-driven spatial random fields with regularly varying L\'evy measure

TL;DR

This paper studies the extreme behavior of Lévy-driven spatial random fields with heavy-tailed Lévy measures, establishing asymptotic tail equivalences and a convergence to Fréchet distribution for the supremum over expanding sets.

Contribution

It provides new asymptotic results for the tail behavior and extreme value distribution of Lévy-driven fields with regularly varying Lévy measures.

Findings

Tail of supremum asymptotically matches Lévy measure tail times kernel integral

Supremum over expanding sets converges to Fréchet distribution

Extreme value behavior characterized for Lévy-driven spatial fields

Abstract

We consider an infinitely divisible random field indexed by $\mathbb{R}^d$, $d\in\mathbb{N}$, given as an integral of a kernel function with respect to a L\'evy basis with a L\'evy 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 L\'evy 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 Fr\'echet distribution.