Extremes of Gaussian processes over an infinite horizon

Abstract

Consider a centered separable Gaussian process Y with a variance function that is regularly varying at infinity with index 2H 2 ð0; 2Þ: Let f be a 'drift' function that is strictly increasing, regularly varying at infinity with index b4H; and vanishing at the origin. Motivated by queueing and risk models, we investigate the asymptotics for u ! 1 of the probability Pðsup tX0 Y t À fðtÞ4uÞ as u ! 1: To obtain the asymptotics, we tailor the celebrated double sum method to our general framework. Two different families of correlation structures are studied, leading to four qualitatively different types of asymptotic behavior. A generalized Pickands' constant appears in one of these cases. Our results cover both processes with stationary increments (including Gaussian integrated processes) and self-similar processes.

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 .…”

On the Dependence Structure of Gaussian Queues

“…[6,11] , whereas exact asymptotics can be found in, e.g., Refs. [10,22] , and the many-sources regime logarithmic asymptotics are in Refs. [1,4] and the exact asymptotics in Ref.…”

Open problems in Gaussian fluid queueing theory

“…Over the past decade, the asymptotic behavior of P(Q > u), as u → ∞, was an important research theme, both for FBM and IG driven queues; see [3,10,13,14]. The structural form of these asymptotics is known now, and captured by the following general formula:…”

“…We note that, for models with {X(t) : t ∈ R} having a regularly varying variance function at ∞ with α ∞ > 1, the asymptotic constant C reduces to the classical Pickands constant H B H with H = α ∞ /2; see [10,13].…”