Extremes of a certain class of Gaussian processes

Hüsler, Jürg; Piterbarg, V. I.

doi:10.1016/s0304-4149(99)00041-1

Cited by 147 publications

(178 citation statements)

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“…Numerous papers have considered the calculation of Pickands constants, with particular focus on the case δ = 0; see for instance Shao (1996); Hüsler and Piterbarg (1999); Dȩbicki et al (2003); Dȩbicki (2005); Dȩbicki and Kisowski (2008); Harper (2013Harper ( , 2015. Recently, the seminal contribution Dieker and Yakir (2014) The principal advantage of Dieker-Yakir representation (3) is that it is given as an expectation rather than as a limit, which is particularly useful for Monte Carlo simulations of H δ W .…”

Section: Introductionmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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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 (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 Lévy processes. We furthermore provide a link to spatial 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.

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Section: Introductionmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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“…Thirdly, although it might appear that the Husler-Piterbarg asymptote [14] also provides adequate accuracy over the full practical range of the parameters (for our purposes, for example, its weakness when H and x are small is not practically important), this formula cannot readily be used independently of simulations because it makes use of a constant for which there is no clear method of calculation. In order to use the Husler-Piterbarg asymptote in the above comparisons, this constant was estimated by simulations as discussed in Section II.…”

Section: Discussion Of Resultsmentioning

confidence: 99%

“…The authors of [14] did not provide a way to computeC. Therefore we estimate this constant by numerically fitting (4) to simulation results.…”

Section: A New Analytical Results Of An Fbm Queuementioning

confidence: 99%

Link dimensioning for fractional Brownian input

Chen

Addie

Zukerman

2011

2011 IEEE 12th International Conference on High Performance Switching and Routing

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Abstract-This paper provides a new formula for dimensioning of a link fed by fractional Brownian input. This formula is obtained based on another new approximate result for the stationary workload distribution of a queue loaded by fractional Brownian input. An efficient approach to simulate such a queue is also presented. Agreement between the analytical and the simulation results has been demonstrated numerically.Index Terms-fractional Brownian noise, fractional Brownian motion, long range dependence, quality of service (QoS), link capacity dimensioning.

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“…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:…”

Section: Estimates and Simulation Of The Asymptotic Constantmentioning

confidence: 99%

“…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].…”

Section: Estimates and Simulation Of The Asymptotic Constantmentioning

confidence: 99%

Open problems in Gaussian fluid queueing theory

Dȩbicki

Mandjes

2011

Queueing Syst

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We present three challenging open problems that originate from the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, we address the problem of characterizing the correlation structure of the stationary buffer content process, the speed of convergence to stationarity, and analysis of an asymptotic constant associated with the stationary buffer content distribution (the so-called Pickands constant).

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Extremes of a certain class of Gaussian processes

Cited by 147 publications

References 1 publication

Generalized Pickands constants and stationary max-stable processes

Generalized Pickands constants and stationary max-stable processes

Link dimensioning for fractional Brownian input

Open problems in Gaussian fluid queueing theory

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