A New Proof of an Old Result by Pickands

where h> 0, α ∈ (0,2] and C > 0 are constants. Note that (1) includes, for example, covariance functions of the form e−|t| α , where the case when α = 1 corresponds to an Ornstein-Uhlenbeck process. Further, the case when α= 2 in (1) corresponds to mean-square differentiable processes, while processes with 0< α < 2 are non-differentiable. The tail distribution of supt∈[0,h] ξ(t) was originally obtained in [6] by means of a long and complicated proof involving so called ε-upcrossings, although his proof was not quite complete. The Pickands result is as follows: