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Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

Part of: Geometric probability and stochastic geometry Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  25 February 2021

Mads Stehr  and
Anders Rønn-Nielsen
Mads Stehr*
Affiliation:
Aarhus University
Anders Rønn-Nielsen*
Affiliation:
Copenhagen Business School
*
*Postal address: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark. Email address: mads.stehr@math.au.dk
**Postal address: Center for Statistics, Department of Finance, Copenhagen Business School, Solbjerg Pl. 3, 2000 Frederiksberg, Denmark.
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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 equivalenceinfinite divisibilityLévy-based modellingasymptotic equivalencesample paths for random fields

MSC classification

Primary: 60G60: Random fields 60G17: Sample path properties
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 42 - 67
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/jpr.2020.73

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