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

  • Anders Rønn-Nielsen and Eva B. Vedel Jensen (a1)
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.

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Corresponding author
*Department of Mathematical Sciences, University of Copenhagen, Universitetspark 5, 2100 Copenhagen Ø, Denmark. Email address: arnielsen@math.ku.dk
References
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[1]Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.
[2]Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
[3]Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2010). Excursion sets of three classes of stable random fields. Adv. Appl. Prob. 42, 293318.
[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, 134169.
[5]Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. 24, 113.
[6]Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 4168.
[7]Barndorff-Nielsen, O. E. (2010). Lévy bases and extended subordination. Res. Rep. 10-12, Department of Mathematics, Aarhus University.
[8]Barndorff-Nielsen, O. E. (2011). Stationary infinitely divisible processes. Braz. J. Prob. Statist. 25, 294322.
[9]Barndorff-Nielsen, O. E. and Schmiegel, J. (2004). Lévy-based spatial-temporal modelling with applications to turbulence. Uspekhi Mat. Nauk. 159, 6390.
[10]Braverman, M. and Samorodnitsky, G. (1995). Functionals of infinitely divisible stochastic processes with exponential tails. Stoch. Process. Appl. 56, 207231.
[11]Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Relat. Fields 72, 529557.
[12]Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. A 43, 347365. (Corrigendum: 48 (1990), 152–153.)
[13]Fasen, V. (2009). Extremes of Lévy driven mixed MA processes with convolution equivalent distributions. Extremes 12, 265296.
[14]Guttorp, P. and Gneiting, T. (2006). Studies of the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika 93, 989995.
[15]Hashorva, E. and Ji, L. (2016). Extremes of α(t)-locally stationary Gaussian random fields. Trans. Amer. Math. Soc. 368, 126.
[16]Hellmund, G., Prokešová, M. and Jensen, E. B. V. (2008). Lévy-based Cox point processes. Adv. Appl. Prob. 40, 603629.
[17]Jónsdóttir, K. Ý., Schmiegel, J. and Vedel Jensen, E. B. (2008). Lévy-based growth models. Bernoulli 14, 6290.
[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, 511529.
[19]Marcus, M. B. and Rosiński, J. (2005). Continuity and boundedness of infinitely divisible processes: a Poisson point process approach. J. Theoret. Prob. 18, 109160.
[20]Maruyama, G. (1970). Infinitely divisible processes. Theory Prob. Appl. 15, 122.
[21]Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407424.
[22]Pedersen, J. (2003). The Lévy–Ito decomposition of an independently scattered random measure. Res. Rep. 2003--2, MaPhySto.
[23]Potthoff, J. (2009). Sample properties of random fields. I. Separability and measurability. Commun. Stoch. Analysis 3, 143153.
[24]Rajput, B. S. and Rosiński, J. (1989). Spectral representations of infinitely divisible processes. Prob. Theory Relat. Fields 82, 451488.
[25]Rønn-Nielsen, A. and Jensen, E. B. V. (2014). Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure. In preparation.
[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.
[27]Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Prob. 21, 9961014.
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