Uniform tail approximation of homogenous functionals of Gaussian fields

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. 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.

Approximation of Sojourn Times of Gaussian Processes

“…The case x = 0, η = 0 is shown in [38]. Since the case x = 0, η > 0 can be established by arguments similar to the presented above, we omit the details.…”

“…We present below an extension of Theorem 2.1 in [38]. Hereafter, C i , i ∈ N are positive constants which might be different from line to line.…”

Sojourn Times of Gaussian Processes with Trend

“…Assumptions (11) and (13) are satisfied for large classes of Gaussian processes, see e.g., [17], [18], [19] and [20]. For example, they are compatible with those in Theorem 3.2 in [21].…”

Extremes of vector-valued Gaussian processes

“…The key to the methodology developed in this contribution is what we refer to as the uniform Pickands-Piterbarg lemma, see Lemma 4.7 below. We briefly explain the main ideas underlying the approach taken in this paper pointing out some subtle issues related to uniform approximations that appear to have been overlooked in the literature; [25] takes particular care of those issues in the one-dimensional setting.…”