Elect. Comm. in Probab. 15 (2010), 339–345 ELECTRONIC COMMUNICATIONS in PROBABILITY A NEW PROOF OF AN OLD RESULT BY PICKANDS

Let {ξ(t)}t∈[0,h] be a stationary Gaussian process with covariance function r such that r(t) = 1 − C |t|α + o(|t|α) as t → 0. We give a new and direct proof of a result originally obtained by Pickands, on the asymptotic behaviour as u→ ∞ of the probability P{supt∈[0,h] ξ(t)> u} that the process ξ exceeds the level u. As a by-product, we obtain a new expression for Pickands constant Hα. 1 Introduction and main result Let {ξ(t)}t∈[0,h] be a continuous centered stationary Gaussian process with covariance function r(t) = Cov{ξ(s),ξ(s+ t)} that satisfies r(t)< 1 for t ∈ (0, h] and r(t) = 1 − C |t|α + o(|t|α) as t → 0, (1) where h> 0, α ∈ (0,2] and C> 0 are constants. Note that (1) includes, for example, cov- arianc