A new proof of an old result by Pickands

Abstract

Let $\{\xi(t)\}_{t\in[0,h]}$ be a stationary Gaussian process with covariance function $r$ such that $r(t) =1-C|t|^{\alpha}+o(|t|^{\alpha})$ as $t\to0$. We give a new and direct proof of a result originally obtained by Pickands, on the asymptotic behaviour as $u\to\infty$ of the probability $\Pr\{\sup_{t\in[0,h]}\xi(t)>u\}$ that the process $\xi$ exceeds the level $u$. As a by-product, we obtain a new expression for Pickands constant $H_\alpha$.