Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T11:06:00.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure

Published online by Cambridge University Press:  24 March 2016

Anders Rønn-Nielsen  and
Eva B. Vedel Jensen
Eva B. Vedel Jensen
Affiliation:
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark. Email address: eva@math.au.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a continuous, infinitely divisible random field in Rd 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 supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Keywords

Asymptotic supremumconvolution equivalenceinfinite divisibilityLévy-based modelling
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 244 - 261
Copyright
Copyright © Applied Probability Trust 2016 

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. (1981). The Geometry of Random Fields. John Wiley, Chichester. Google Scholar
[2]Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. Google Scholar
[3]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
[4]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. Google Scholar
[5]Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13. Google Scholar
[6]Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68. Google Scholar
[7]Barndorff-Nielsen, O. E. (2010). Lévy bases and extended subordination. Res. Rep. 10-12, Department of Mathematics, Aarhus University. Google Scholar
[8]Barndorff-Nielsen, O. E. (2011). Stationary infinitely divisible processes. Braz. J. Prob. Statist. 25, 294–322. Google Scholar
[9]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
[10]Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207–231. Google Scholar
[11]Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557. Google Scholar
[12]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.) Google Scholar
[13]Fasen, V. (2009). Extremes of Lévy driven mixed MA processes with convolution equivalent distributions. Extremes 12, 265–296. Google Scholar
[14]Guttorp, P. and Gneiting, T. (2006). Studies of the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93, 989–995. Google Scholar
[15]Hashorva, E. and Ji, L. (2016). Extremes of α(t)-locally stationary Gaussian random fields. Trans. Amer. Math. Soc. 368, 1–26. Google Scholar
[16]Hellmund, G., Prokešová, M. and Jensen, E. B. V. (2008). Lévy-based Cox point processes. Adv. Appl. Prob. 40, 603–629. Google Scholar
[17]Jónsdóttir, K. Ý., Schmiegel, J. and Vedel Jensen, E. B. (2008). Lévy-based growth models. Bernoulli 14, 62–90. Google Scholar
[18]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
[19]Marcus, M. B. and Rosiński, J. (2005). Continuity and boundedness of infinitely divisible processes: a Poisson point process approach. J. Theoret. Prob. 18, 109–160. Google Scholar
[20]Maruyama, G. (1970). Infinitely divisible processes. Theory Prob. Appl. 15, 1–22. Google Scholar
[21]Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424. Google Scholar
[22]Pedersen, J. (2003). The Lévy–Ito decomposition of an independently scattered random measure. Res. Rep. 2003--2, MaPhySto. Google Scholar
[23]Potthoff, J. (2009). Sample properties of random fields. I. Separability and measurability. Commun. Stoch. Analysis 3, 143–153. Google Scholar
[24]Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–488. Google Scholar
[25]Rønn-Nielsen, A. and Jensen, E. B. V. (2014). Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure. In preparation. Google Scholar
[26]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
[27]Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014. Google Scholar