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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.
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Abstract

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 ξu(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,τu(t), t ∈ ℝd. In this paper we derive uniform approximations of ℙ{Γ(ξu,τu) > u} with respect to τu, in some index set Ku 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
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1037 - 1066
Copyright
Copyright © Applied Probability Trust 2017 

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