Journal of Applied Probability

Extremes of homogeneous Gaussian random fields

Krzysztof Dębicki, Enkelejd Hashorva, and Natalia Soja-Kukieła

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let {X(s, t): s, t ≥ 0} be a centred homogeneous Gaussian field with almost surely continuous sample paths and correlation function r(s, t) = cov(X(s, t), X(0, 0)) such that r(s, t) = 1 - |s|α1 - |t|α2 + o(|s|α1 + |t|α2), s, t → 0, with α1, α2 ∈ (0, 2], and r(s, t) < 1 for (s, t) ≠ (0, 0). In this contribution we derive an asymptotic expansion (as u → ∞) of P(sup(sn1(u),tn2(u)) ∈[0,x]∙[0,y]X(s, t) ≤ u), where n1(u)n2(u) = u2/α1+2/α2Ψ(u), which holds uniformly for (x, y) ∈ [A, B]2 with A, B two positive constants and Ψ the survival function of an N(0, 1) random variable. We apply our findings to the analysis of extremes of homogeneous Gaussian fields over more complex parameter sets and a ball of random radius. Additionally, we determine the extremal index of the discretised random field determined by X(s, t).

Article information

Source
J. Appl. Probab. Volume 52, Number 1 (2015), 55-67.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1429282606

Digital Object Identifier
doi:10.1239/jap/1429282606

Mathematical Reviews number (MathSciNet)
MR3336846

Zentralblatt MATH identifier
1315.60059

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes

Keywords
Gaussian random field supremum tail asymptoticy extremal index Berman condition strong dependence

Citation

Dębicki, Krzysztof; Hashorva, Enkelejd; Soja-Kukieła, Natalia. Extremes of homogeneous Gaussian random fields. J. Appl. Probab. 52 (2015), no. 1, 55--67. doi:10.1239/jap/1429282606. https://projecteuclid.org/euclid.jap/1429282606


Export citation

References

  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Arendarczyk, M. and Dębicki, K. (2012). Exact asymptotics of supremum of a stationary Gaussian process over a random interval. Statist. Prob. Lett. 82, 645–652.
  • Tan, Z. and Hashorva, E. (2013). Limit theorems for extremes of strongly dependent cyclo-stationary $\chi$-processes. Etremes 16, 241–254.
  • Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73.
  • Piterbarg, V. I. (1972). On the paper by J. Pickands `Upcrossing probabilities for stationary Gaussian processes'. Vestnik Moskov. Univ. Ser. I Mat. Meh. 27, 25–30.
  • Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields ( Trans. Math. Monogr. 148). American Mathematical Society, Providence, RI.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
  • Jakubowski, A. (1991). Relative extremal index of two stationary processes. Stoch. Process. Appl. 37, 281–297.
  • Leadbetter, M. R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrscheinlichkeitsch. 65, 291–306.
  • O'Brien, G. L. (1987). Extreme values for stationary and Markov sequences. Ann. Prob. 15, 281–291.
  • French, J. P. and Davis, R. A. (2013). The asymptotic distribution of the maxima of a Gaussian random field on a lattice. Extremes 16, 1–26.
  • Hsing, T. (1993). Extremal index estimation for a weakly dependent stationary sequence. Ann. Statist. 21, 2043–2071.
  • Jakubowski, A. and Soja-Kukieła, N. (2014). Managing local dependencies in limit theorems for maxima of stationary random fields. Submitted..
  • Ferreira, H. (2006). The upcrossings index and the extremal index. J. Appl. Prob. 43, 927–937.
  • Laurini, F. and Tawn, J. A. (2012). The extremal index for GARCH$(1,1)$ processes. Extremes 15, 511–529.
  • Dębicki, K., Hashorva, E. and Soja-Kukieła, N. (2013). Extremes of homogeneous Gaussian random fields. Preprint. Available at http://arxiv.org/abs/1312.2863.