A new proof of an old result by pickands

Let (ξ(t))_{t ∈[o, 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(sup_{t ∈ [0, h]} ξ (t) > u) that the process > exceeds the level u. As a by-product, we obtain a new expression for Pickands constant _α.