Uniform tail approximation of homogenous functionals of Gaussian fields. (17th November 2017) - Dȩbicki, Krzysztof; Hashorva, Enkelejd; Liu, Peng
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 ℙΓ(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 ℙΓ(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.