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Extremes

, Volume 20, Issue 2, pp 333–392 | Cite as

Extremes of Gaussian random fields with regularly varying dependence structure

  • Krzysztof Dȩbicki
  • Enkelejd Hashorva
  • Peng Liu
Article

Abstract

Let \(X(t), t\in \mathcal {T}\) be a centered Gaussian random field with variance function σ 2(⋅) that attains its maximum at the unique point \(t_{0}\in \mathcal {T}\), and let \(M(\mathcal {T})=\sup _{t\in \mathcal {T}} X(t)\). For \(\mathcal {T}\) a compact subset of ℝ, the current literature explains the asymptotic tail behaviour of \(M(\mathcal {T})\) under some regularity conditions including that 1 − σ(t) has a polynomial decrease to 0 as tt 0. In this contribution we consider more general case that 1 − σ(t) is regularly varying at t 0. We extend our analysis to Gaussian random fields defined on some compact set \(\mathcal {T}\subset \mathbb {R}^{2}\), deriving the exact tail asymptotics of \(M(\mathcal {T})\) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0. A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

Keywords

Non-stationary Gaussian processes Gaussian random fields Extremes Fractional Brownian motion Regular variation Uniform approximation 

AMS 2000 Subject Classifications

Primary 60G15 secondary 60G70 

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References

  1. Adler, R., Taylor, J.: Random fields and geometry. Springer Monographs in Mathematics. Springer, New York (2007)Google Scholar
  2. Arendarczyk, M.: On the asymptotics of supremum distribution for some iterated processes, Extremes. doi: 10.1007/s10687-016-0272-2 (2016)
  3. Azaïs, J., Wschebor, M.: Level sets and extrema of random processes and fields. Wiley, Hoboken, NJ (2009)CrossRefzbMATHGoogle Scholar
  4. Azmoodeh, E., Sottinen, T., Viitasaari, L., Yazigi, A.: Necessary and sufficient conditions for Hölder continuity of Gaussian processes. Statist. Probab. Lett. 94, 230–235 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Berman, S.: Sojourns and extremes of stochastic processes. The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Grove, CA (1992)Google Scholar
  6. Bingham, N., Goldie, C., Teugels, J.: Regular variation, vol. 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  7. Burnecki, K., Michna, Z.: Simulation of Pickands constants. Probab. Math. Statist. 22(1) (2002). Acta Univ. Wratislav. No. 2409, pp. 193–199Google Scholar
  8. Cheng, D., Schwartzman, A.: Distribution of the height of local maxima of Gaussian random fields. Extremes 18(2), 213–240 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cheng, D., Xiao, Y.: The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments. Annals Appl. Probab. (2016). in pressGoogle Scholar
  10. Cheng, D.: Excursion probabilities of isotropic and locally isotropic Gaussian random fields on manifolds, Extremes. doi: 10.1007/s10687-016-0271-3(2016)
  11. Dȩbicki, K.: Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98(1), 151–174 (2002)MathSciNetCrossRefGoogle Scholar
  12. Dieker, A.B.: Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115(2), 207–248 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dȩbicki, K., Kosiński, K.: On the infimum attained by the reflected fractional Brownian motion. Extremes 17(3), 431–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Dieker, A.B., Yakir, B.: On asymptotic constants in the theory of Gaussian processes. Bernoulli 20(3), 1600–1619 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Dȩlhkebicki, K., Hashorva, E., Ji, L.: Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals. Extremes 17 (3), 411–429 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dȩbicki, K., Hashorva, E., Ji, L., Tabiś, K.: Extremes of vector-valued Gaussian processes: Exact asymptotics. Stoch. Process. Appl. 125(11), 4039–4065 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Dȩbicki, K., Kisowski, P.: Asymptotics of supremum distribution of α(t)-locally stationary Gaussian processes. Stoch. Process. Appl. 118(11), 2022–2037 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Dȩbicki, K., Hashorva, E., Ji, L.: Extremes of a class of nonhomogeneous Gaussian random fields. Ann. Probab. 44(2), 984–1012 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Dȩbicki, K., Hashorva, E., Liu, P.: Uniform tail approximation of homogenous functionals of Gaussian fields, arXiv:1607.01430.pdf (2016)
  20. Dieker, A.B., Mikosch, T.: Exact simulation of Brown-Resnick random fields at a finite number of locations. Extremes 18, 301–314 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Dȩbicki, K., Engelke, S., Hashorva, E.: Brown-Resnick processes and Pickands-type constants, arXiv:1602.01613.pdf (2016)
  22. Dȩbicki, K., Hashorva, E., Soja-Kukieła, N.: Extremes of homogeneous Gaussian random fields. J. Appl. Probab. 52(1), 55–67 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Du, Y., Miao, J., Wu, D., Xiao, Y.: Packing dimensions of the images of Gaussian random fields. Statist. Probab. Lett. 106, 209–217 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling extremal events, vol. 33 of Applications of Mathematics (New York). Springer-Verlag, Berlin (1997)CrossRefzbMATHGoogle Scholar
  25. Farkas, J., Hashorva, E.: Tail approximation for reinsurance portfolios of Gaussian-like risks. Scand. Actuar. J. 4, 319–331 (2015)MathSciNetCrossRefGoogle Scholar
  26. Girard, S., Stupfler, G.: Extreme geometric quantiles in a multivariate regular variation framework. Extremes 18(4), 629–663 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Hashorva, E., Ji, L.: Extremes of α(t)-locally stationary Gaussian random fields. Trans. Amer. Math. Soc. 368(1), 1–26 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Hüsler, J., Piterbarg, V.I.: Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83(2), 257–271 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Hüsler, J., Piterbarg, V.I.: On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113(2), 315–332 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Li, Y., Wang, W., Xiao, Y.: Exact moduli of continuity for operator-scaling Gaussian random fields. Bernoulli 21(2), 930–956 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Michna, Z.: On tail probabilities and first passage times for fractional Brownian motion. Math. Methods Oper. Res. 49(2), 335–354 (1999)MathSciNetzbMATHGoogle Scholar
  32. Pickands, III. J.: Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Piterbarg, V.I.: On the paper by J. Pickands Upcrossing probabilities for stationary Gaussian processes. Vestnik Moskov. Univ. Ser. I Mat. Meh. 27(5), 25-30 (1972)MathSciNetzbMATHGoogle Scholar
  34. Piterbarg, V.I.: Asymptotic methods in the theory of Gaussian processes and fields, vol. 148 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1996)Google Scholar
  35. Piterbarg, V.I.: Twenty lectures about gaussian processes. Atlantic Financial Press, London, New York (2015)zbMATHGoogle Scholar
  36. Piterbarg, V.I.: High extrema of Gaussian chaos processes. Extremes 19(2), 253–272 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Piterbarg, V.I., Prisjažnjuk, V.P.: Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process. Teor. Verojatnost. Mat. Statist. 18, 121–134, 183 (1978)MathSciNetzbMATHGoogle Scholar
  38. Popivoda, G., Stamatovic, S.: Extremes of Gaussian fields with a smooth random variance. Statist. Probab. Lett. 110, 185–190 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  39. Qiao, W., Polonik, W.: Extrema of locally stationary Gaussian elds on growing manifolds, Bernoulli to appear (2016)Google Scholar
  40. Qualls, C., Watanabe, H.: Asymptotic properties of Gaussian processes. Ann. Math. Statist. 43, 580–596 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  41. Resnick, S.: Heavy-tail phenomena. Springer Series in Operations Research and Financial Engineering. Probabilistic and Statistical Modeling, New York (2007)Google Scholar
  42. Samorodnitsky, G.: Continuity of Gaussian processes. Ann. Probab. 16(3), 1019–1033 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  43. Samorodnitsky, G.: Probability tails of Gaussian extrema. Stoch. Process. Appl. 38(1), 55–84 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  44. Samorodnitsky, G., Taqqu, M.: Stochastic monotonicity and Slepian-type inequalities for infinitely divisible and stable random vectors. Ann. Probab. 21(1), 143–160 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  45. Soulier, P.: Some applications of regular variation in probability and statistics. Instituto Venezolano de Investigaciones Cientcas: XXII ESCUELA VENEZOLANA DE MATEMATICAS (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
  • Enkelejd Hashorva
    • 2
  • Peng Liu
    • 1
    • 2
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of LausanneUNIL-DorignySwitzerland

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