Skip to main content

Uniform tail approximation of homogenous functionals of Gaussian fields

  • Krzysztof Dȩbicki (a1), Enkelejd Hashorva (a2) and Peng Liu (a3)

Let X(t), t ∈ ℝ d , be a centered Gaussian random field with continuous trajectories and set ξ u (t) = X(f(u)t), t ∈ ℝ d , with f some positive function. Using classical results we can establish the tail asymptotics of ℙ{Γ(ξ u ) > u} as u → ∞ with Γ(ξ u ) = sup t ∈ [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 τ u K say ξ u u (t), t ∈ ℝ d . In this paper we derive uniform approximations of ℙ{Γ(ξ u u ) > u} with respect to τ u , 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 ξ u . Finally, we show the finiteness of generalized Piterbarg constants.

Corresponding author
* Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address:
** Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland.
*** Email address:
**** Email address:
Hide All
[1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
[2] Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.
[3] Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Prob. 10, 146.
[4] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Pacific Grove, CA.
[5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. App. 27). Cambridge University Press.
[6] Dębicki, K. and Hashorva, E. (2017). On extremal index of max-stable stationary processes. To appear in Prob. Math. Statist. Available at
[7] Dębicki, K. and Kosiński, K. M. (2014). On the infimum attained by the reflected fractional Brownian motion. Extremes 17, 431446.
[8] Dębicki, K. and Liu, P. (2016). Extremes of stationary Gaussian storage models. Extremes 19, 273302.
[9] Dębicki, K. and Mandjes, M. (2003). Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Prob. 40, 704720.
[10] Dębicki, K., Engelke, S. and Hashorva, E. (2017). Generalized Pickands constants and stationary max-stable processes. Extremes 20, 493517
[11] Dębicki, K., Hashorva, E. and Ji, L. (2015). Parisian ruin of self-similar Gaussian risk processes. J. Appl. Prob. 52, 688702.
[12] Dębicki, K., Hashorva, E. and Ji, L. (2016). Extremes of a class of nonhomogeneous Gaussian random fields. Ann. Prob. 44, 9841012.
[13] Dębicki, K., Hashorva, E. and Ji, L. (2016). Parisian ruin over a finite-time horizon. Sci. China Math. 59, 557572.
[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
[15] Dębicki, K., Hashorva, E. and Liu, P. (2017). Extremes of Gaussian random fields with regularly varying dependence structure. Extremes 20, 333392.
[16] Dębicki, K., Kosiński, K. M., Mandjes, M. and Rolski, T. (2010). Extremes of multidimensional Gaussian processes. Stoch. Process. Appl. 120, 22892301.
[17] Dębicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151174.
[18] Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207248.
[19] Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes 18, 301314.
[20] Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 16001619.
[21] Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257271.
[22] Hüsler, J. and Piterbarg, V. (2004). On the ruin probability for physical fractional Brownian motion. Stoch. Process. Appl. 113, 315332.
[23] Kabluchko, Z. (2010). Stationary systems of Gaussian processes. Ann. Appl. Prob. 20, 22952317.
[24] Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 5173.
[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, 2530.
[26] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Transl. Math. Monog. 148). American Mathematical Society, Providence, RI.
[27] Piterbarg, V. I. (2015). Twenty Lectures About Gaussian Processes. Atlantic Financial Press, London.
[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, 171172.
[29] Zhou, Y. and Xiao, Y. (2017). Tail asymptotics for the extremes of bivariate Gaussian random fields. Bernoulli 23, 15661598.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed