Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-10T20:35:22.109Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 833 - 851
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. Google Scholar
[2] Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2010). Excursion sets of three classes of stable random fields. Adv. Appl. Prob. 42, 293–318. Google Scholar
[3] Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2013). High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Prob. 41, 134–169. CrossRefGoogle Scholar
[4] Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). Lévy-based spatial-temporal modelling, with applications to turbulence. Uspekhi Mat. Nauk 159, 63–90. Google Scholar
[5] Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557. CrossRefGoogle Scholar
[6] Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A 43, 347–365. (Corrigendum: 48 (1990), 152–153.) CrossRefGoogle Scholar
[7] Hellmund, G., Prokešová, M. and Jensen, E. B. V. (2008). Lévy-based Cox point processes. Adv. Appl. Prob. 40, 603–629. Google Scholar
[8] Jónsdóttir, K. Ý., Schmiegel, J. and Vedel Jensen, E. B. (2008). Lévy-based growth models. Bernoulli 14, 62–90. Google Scholar
[9] Jónsdóttir, K. Ý., Rønn-Nielsen, A., Mouridsen, K. and Jensen, E. B. V. (2013). Lévy-based modelling in brain imaging. Scand. J. Statist. 40, 511–529. Google Scholar
[10] Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424. Google Scholar
[11] Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014. Google Scholar
[12] Rønn-Nielsen, A. and Jensen, E. B. V. (2014). Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure. Res. Rep. 2014-09, CSGB. Google Scholar
[13] Rønn-Nielsen, A. and Jensen, E. B. V. (2016). Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure. J. Appl. Prob. 53, 244–261. Google Scholar