Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T11:08:51.592Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform tail approximation of homogenous functionals of Gaussian fields

Part of:Stochastic processes

Published online by Cambridge University Press:  17 November 2017

Krzysztof Dȩbicki,
Enkelejd Hashorva and
Peng Liu
Krzysztof Dȩbicki*
Affiliation:
University of Wrocław
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Peng Liu*
Affiliation:
University of Lausanne and University of Wrocław
*
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: krzysztof.debicki@math.uni.wroc.pl
** Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland.
** Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let X(t), t ∈ ℝd , be a centered Gaussian random field with continuous trajectories and set ξu (t) = X(f(u)t), t ∈ ℝd , with f some positive function. Using classical results we can establish the tail asymptotics of ℙ{Γ(ξu ) > u} as u → ∞ with Γ(ξu ) = supt ∈ [0, T] d ξ u(t), T > 0, by requiring that f(u) tends to 0 as u → ∞ with speed controlled by the local behavior of the correlation function of X. Recent research shows that for applications, more general functionals than the supremum should be considered and the Gaussian field can depend also on some additional parameter τu ∈ K say ξu,τ u (t), t ∈ ℝd . In this paper we derive uniform approximations of ℙ{Γ(ξu,τ u ) > u} with respect to τu , in some index set K u as u → ∞. Our main result has important theoretical implications; two applications are already included in Dȩbicki et al. (2016), (2017). In this paper we present three additional applications. First we derive uniform upper bounds for the probability of double maxima. Second, we extend the Piterbarg–Prisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields ξu . Finally, we show the finiteness of generalized Piterbarg constants.

Keywords

Fractional Brownian motionsupremum of Gaussian random fieldsstationary processdouble maximauniform double-sum methodgeneralized Piterbarg constant

MSC classification

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

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017, pp. 1037 - 1066
Copyright
Copyright © Applied Probability Trust 2017 

References

[1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. Google Scholar
[2] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ. Google Scholar
[3] Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46. Google Scholar
[4] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Pacific Grove, CA. CrossRefGoogle Scholar
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. App. 27). Cambridge University Press. Google Scholar
[6] Dębicki, K. and Hashorva, E. (2017). On extremal index of max-stable stationary processes. To appear in Prob. Math. Statist. Available at https://arxiv.org/abs/1704.01563/. Google Scholar
[7] Dębicki, K. and Kosiński, K. M. (2014). On the infimum attained by the reflected fractional Brownian motion. Extremes 17, 431–446. Google Scholar
[8] Dębicki, K. and Liu, P. (2016). Extremes of stationary Gaussian storage models. Extremes 19, 273–302. Google Scholar
[9] Dębicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704–720. Google Scholar
[10] Dębicki, K., Engelke, S. and Hashorva, E. (2017). Generalized Pickands constants and stationary max-stable processes. Extremes 20, 493–517 Google Scholar
[11] Dębicki, K., Hashorva, E. and Ji, L. (2015). Parisian ruin of self-similar Gaussian risk processes. J. Appl. Prob. 52, 688–702. Google Scholar
[12] Dębicki, K., Hashorva, E. and Ji, L. (2016). Extremes of a class of nonhomogeneous Gaussian random fields. Ann. Prob. 44, 984–1012. Google Scholar
[13] Dębicki, K., Hashorva, E. and Ji, L. (2016). Parisian ruin over a finite-time horizon. Sci. China Math. 59, 557–572. Google Scholar
[14] Dębicki, K., Hashorva, E. and Liu, P. (2015). Ruin probabilities and passage times of γ-reflected Gaussian processes with stationary increments. Preprint. Available at https://arxiv.org/abs/1511.09234v1. Google Scholar
[15] Dębicki, K., Hashorva, E. and Liu, P. (2017). Extremes of Gaussian random fields with regularly varying dependence structure. Extremes 20, 333–392. Google Scholar
[16] Dębicki, K., Kosiński, K. M., Mandjes, M. and Rolski, T. (2010). Extremes of multidimensional Gaussian processes. Stoch. Process. Appl. 120, 2289–2301. Google Scholar
[17] Dębicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174. Google Scholar
[18] Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248. Google Scholar
[19] Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes 18, 301–314. Google Scholar
[20] Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 1600–1619. Google Scholar
[21] Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271. Google Scholar
[22] Hüsler, J. and Piterbarg, V. (2004). On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113, 315–332. Google Scholar
[23] Kabluchko, Z. (2010). Stationary systems of Gaussian processes. Ann. Appl. Prob. 20, 2295–2317. Google Scholar
[24] Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73. Google Scholar
[25] Piterbarg, V. I. (1972). On the paper by J. Pickands 'Upcrosssing probabilities for stationary Gaussian processes'. Vestnik Moscow Univ Ser. I Mat. Meh. 27, 25–30. Google Scholar
[26] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monog. 148). American Mathematical Society, Providence, RI. Google Scholar
[27] Piterbarg, V. I. (2015). Twenty Lectures About Gaussian Processes. Atlantic Financial Press, London. Google Scholar
[28] Piterbarg, V. I. and Stamatovich, B. (2005). Rough asymptotics of the probability of simultaneous high extrema of two Gaussian processes: the dual action functional. Uspekhi Mat. Nauk 60, 171–172. Google Scholar
[29] Zhou, Y. and Xiao, Y. (2017). Tail asymptotics for the extremes of bivariate Gaussian random fields. Bernoulli 23, 1566–1598. CrossRefGoogle Scholar