Uniform tail approximation of homogenous functionals of Gaussian fields. (17th November 2017)

• Record Type: Journal Article
• Authors: Dȩbicki, Krzysztof; Hashorva, Enkelejd; Liu, Peng
• Is Part Of: Advances in applied probability. Volume 49:Number 4(2017)
• Page Start: 1037
• Page End: 1066
• Publication Date: 2017-11-17
• Subjects: Fractional Brownian motion, -- supremum of Gaussian random fields, -- stationary process, -- double maxima, -- uniform double-sum method, -- generalized Piterbarg constant; Primary 60G15, -- Secondary 60G70; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2017.33
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: 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, τ u ( t ), t ∈ ℝ d . In this paper we derive uniform approximations of ℙ{Γ(ξ u, τ 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.