REMARKS ON PICKANDS’ THEOREM
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.