Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-25T06:53:28.623Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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