Remarks on Pickands theorem

TL;DR

This paper provides a complete proof of Pickands theorem using Borell inequality and Slepian lemma, simplifying the original complex proof and offering a lower bound for Pickands constant.

Contribution

It presents a simplified, self-contained proof of Pickands theorem based on general lemmas, and includes a new lower bound for Pickands constant.

Findings

Complete proof of Pickands theorem using Borell inequality and Slepian lemma

Simplification of the original proof structure

Establishment of a lower bound for Pickands constant

Abstract

In this article we present 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 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.