Extremes of stationary Gaussian storage models

Abstract

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.

An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

“…Norros (2004). In particular, in the seminal paper by Hüsler and Piterbarg (1999) the exact asymptotics of one dimensional marginal distributions of Q B H was derived; see also Dieker (2005), Dębicki (2002), and Dębicki and Liu (2016) for results on more general Gaussian input processes. The purpose of this paper is to investigate the asymptotic 0-1 behavior of the processes Q B H .…”

Sample path properties of reflected Gaussian processes

“…where the remaining constants are defined in Equation 3.43. Since the exact asymptotics of ψ(u), as u grows large, were found in (Dębicki and Liu, 2016), c.f., Theorem 3.1, it follows that ψ(f p (u)) f p (u) = C (u log 1−p u) −1 (1 + o(u)), as u → ∞.…”

Extremes ofγ-reflected Gaussian processes with stationary increments

“…lim u→∞ uσ 2 (u) σ 2 (u) = 2α ∞ (2α ∞ − 1). (52) Lemma 5.2 in [46] shows that AI implies that in a neighborhood of 0…”

“…with respect to τ ∈ K u . We need the following assumptions, which are similar to those imposed in [46] There exists a centered Gaussian random field V (t), t ∈ R d with V (0) = 0, covariance function (σ 2…”