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Tail Asymptotics of an Infinitely Divisible Space-Time Model with Convolution Equivalent Lévy Measure

By Mads Stehr and Anders Rønn-Nielsen
CSGB Research Reports
No. 09, September 2019
Abstract:

We consider a space-time random field on $\mathbb{R}^d \times \mathbb{R}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time-axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time-point and a rotation of a spatial object with fixed radius, in which the field exceeds the level $x$, and that there is a time-interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level $x$.

Keywords: Convolution equivalence; infinite divisibility; Lévy-based modelling; asymptotic equivalence; sample paths for random fields

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