Krzysztof Dębicki

TL;DR: In this article, the authors provide an extensive introduction to queueing models driven by Levy-processes as well as a systematic account of the literature on Levy-driven queues, making the reader familiar with the wide set of probabilistic techniques that have been developed over the past decades, including transform-based techniques, martingales, rateconservation arguments, change-of-measure, importance sampling and large deviations.

TL;DR: This paper considers a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments and considers both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon.

TL;DR: In this article, the exact asymptotics of the supremum distribution of fractional Laplace motion were derived for a centered Gaussian process with stationary increments and variance function.

TL;DR: In this paper, the authors derived the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes with respect to a non-negative constant.

Abstract: Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (Bernoulli, 20(3), 1600–1619, 2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker–Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Levy processes. We furthermore develop a link to extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.