Uniform tail approximation of homogenous functionals of Gaussian fields

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.