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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 37, Fasc. 2,
pages 373 - 393
 

REMARKS ON PICKANDS’ THEOREM

Zbigniew Michna

Abstract: 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.

2010 AMS Mathematics Subject Classification: Primary: 60G15; Secondary: 60G70.

Keywords and phrases: Stationary Gaussian process, supremum of a process, Pickands constant, fractional Brownian motion.

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