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Pickands–Piterbarg Constants for Self-Similar Gaussian Processes

Dębicki Krzysztof, Tabiś Kamil

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

297--315

For a centered self-similar Gaussian process {Y (t) : t ∈ [0;∞)} and R≥0 we analyze the asymptotic behavior of [formula], for suitably chosen γ> 0. Additionally, we find bounds for HRY , R > 0, and a surprising relation between HY and the classical Pickands constants.

Remarks on Pickands’ theorem

Michna Z.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

373--393

In this article we present the Pickands theorem and his double sum method. We follow Piterbarg’s proof of this theorem. Since his proof relies on general lemmas, we present a complete proof of Pickands’ theorem using the Borell inequality and Slepian lemma. The original Pickands’ proof is rather complicated and is mixed with upcrossing probabilities for stationary Gaussian processes. We give a lower bound for Pickands constant. Moreover, we review equivalent definitions, simulations and bounds of Pickands constant.

On extremal index of max-stable stationary processes

Dębicki K., Hashorva E.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

299--317

In this contribution we discuss the relation between Pickands-type constants defined for certain Brown-Resnick stationary proces W(t), t ϵ R, as [wzór] (set 0Z = R if δ = 0) and the extremal index of the associated max-stable stationary process ξW. We derive several new formulas and obtain lower bounds for ΉδW if W is a Gaussian or a Lévy process. As a by-product we show an interesting relation between Pickands constants and lower tail probabilities for fractional Brownian motions.

Extremes of chi-square processes with trend

Liu P., Ji L.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 1

1--20

This paper studies the supremum of chi-square processes with trend over a threshold-dependent-time horizon. Under the assumptions that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square proces with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.

Simulation of Pickands constants

Burnecki K., Michna Z.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

193--199

Pickands constants appear in the asymptotic formulas for extremes of Gaussian processes. The explicit formula of Pickands constants does not exist. Moreover, in the literature there is no numerical approximation. In this paper we compute numerically Pickands constants by the use of change of measure technique. To this end we apply two different algorithms to simulate fractional Brownian motion. Finally, we compare the approximations with a theoretical hypothesis and a recently obtained lower bound on the constants. The results justify the hypothesis.