## Journal of Applied Probability

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

#### 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.

#### Article information

Source
J. Appl. Probab., Volume 53, Number 1 (2016), 244-261.

Dates
First available in Project Euclid: 8 March 2016

https://projecteuclid.org/euclid.jap/1457470572

Mathematical Reviews number (MathSciNet)
MR3471960

Zentralblatt MATH identifier
1337.60032

#### Citation

Rønn-Nielsen, Anders; Jensen, Eva B. Vedel. Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure. J. Appl. Probab. 53 (2016), no. 1, 244--261. https://projecteuclid.org/euclid.jap/1457470572

#### References

• Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.
• Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
• 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.
• 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.
• Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. 24, 1–13.
• Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.
• Barndorff-Nielsen, O. E. (2010). Lévy bases and extended subordination. Res. Rep. 10-12, Department of Mathematics, Aarhus University.
• Barndorff-Nielsen, O. E. (2011). Stationary infinitely divisible processes. Braz. J. Prob. Statist. 25, 294–322.
• Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). Lévy-based spatial-temporal modelling with applications to turbulence. Uspekhi Mat. Nauk. 159, 63–90.
• Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207–231.
• Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529–557.
• 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.)
• Fasen, V. (2009). Extremes of Lévy driven mixed MA processes with convolution equivalent distributions. Extremes 12, 265–296.
• Guttorp, P. and Gneiting, T. (2006). Studies of the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93, 989–995.
• Hashorva, E. and Ji, L. (2016). Extremes of $\alpha(t)$-locally stationary Gaussian random fields. Trans. Amer. Math. Soc. 368, 1–26.
• Hellmund, G., Prokešová, M. and Jensen, E. B. V. (2008). Lévy-based Cox point processes. Adv. Appl. Prob. 40, 603–629.
• Jónsdóttir, K. \'Y., Schmiegel, J. and Vedel Jensen, E. B. (2008). Lévy-based growth models. Bernoulli 14, 62–90.
• Jónsdóttir, K. \'Y., Rønn-Nielsen, A., Mouridsen, K. and Jensen, E. B. V. (2013). Lévy-based modelling in brain imaging. Scand. J. Statist. 40, 511–529.
• 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.
• Maruyama, G. (1970). Infinitely divisible processes. Theory Prob. Appl. 15, 1–22.
• Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.
• Pedersen, J. (2003). The Lévy–Ito decomposition of an independently scattered random measure. Res. Rep. 2003–2, MaPhySto.
• Potthoff, J. (2009). Sample properties of random fields. I. Separability and measurability. Commun. Stoch. Analysis 3, 143–153.
• Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451–488.
• Rønn-Nielsen, A. and Jensen, E. B. V. (2014). Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure. In preparation.
• 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.
• Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 996–1014.