Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

Abstract

We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure

“…Also, this probability is asymptotically described by the right tail of the Lévy measure. The proof is based on the same type of reasoning as in Sections 3 and 4 and is part of a forthcoming paper [25].…”

Central limit theorem for mean and variogram estimators in Lévy–based models

“…Due to the tractability of Lévy-based models, it has been possible to derive tail asymptotics for the supremum of such a field as well as asymptotics of excursion sets of the field. The case where M is a convolution equivalent measure is considered in [12] and [13]. Results for random measures with regularly varying tails have been derived in [15], and refined in [1] and [2].…”