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

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

Published online by Cambridge University Press:  15 September 2017

Anders Rønn-Nielsen  and
Eva B. Vedel Jensen
Anders Rønn-Nielsen*
Affiliation:
Copenhagen Business School
Eva B. Vedel Jensen*
Affiliation:
Aarhus University
*
* Postal address: Department of Finance, Copenhagen Business School, Solbjerg Plads 3, DK-2000 Frederiksberg, Denmark. Email address: aro.fi@cbs.dk
** Postal address: Department of Mathematics and Centre for Stochastic Geometry and Advanced Bioimaging, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark. Email address: eva@imf.au.dk
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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.

Keywords

Convolution equivalenceexcursion setinfinite divisibilityLévy-based modelling

MSC classification

Primary: 60G60: Random fields
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 833 - 851
Copyright
Copyright © Applied Probability Trust 2017 

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