is a centered Gaussian process with stationary increments, c > 0 and β > 0 is chosen such that Q(t) is finite a.s., we derive exact asymptotics of P sup t∈[0,T u ] Q(t) > u and P inf t∈[0,T u ] Q(t) > u , as u → ∞. As a by-product we find conditions under which strong Piterbarg property holds.

Let X(t), t ∈ T be a centered Gaussian random field with variance function σ 2 (·) that attains its maximum at the unique point t 0 ∈ T , and let M (T ) := sup t∈T X(t). For T a compact subset of R, the current literature explains the asymptotic tail behaviour of M (T ) under some regularity conditions including that 1 − σ(t) has a polynomial decrease to 0 as t → t 0 . In this contribution we consider more general case that 1 − σ(t) is regularly varying at t 0 . We extend our analysis to random fields defined on some compact T ⊂ R 2 , deriving the exact tail asymptotics of M (T ) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0 . A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

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. Classical results establish the tail asymptotics of P {Γ(ξ u ) > u}, T > 0 by requiring that f (u) tends to 0 as u → ∞ with speed controlled by the local behaviour of the correlation function of X. Recent research shows that for applications more general functionals than supremum should be considered and the Gaussian field can depend also on some additional parameter τ u ∈ K, say ξ u,τu (t), t ∈ R d . In this contribution we derive uniform approximations of P {Γ(ξ u,τu ) > u} with respect to τ u in some index set K u , as u → ∞. Our main result have important theoretical implications; two applications are already included in [12,13]. In this paper we present three additional ones, namely i) we derive uniform upper bounds for the probability of double-maxima, ii) we extend PiterbargPrisyazhnyuk theorem to some large classes of homogeneous functionals of centered Gaussian fields ξ u , and iii)we show the finiteness of generalized Piterbarg constants.