Upcrossing probabilities for stationary Gaussian processes

Abstract

JAMES PICKANDS IIIO 1. Introduction.Let {X(t), -co 0, where X'(t0) is the derivative of the realization X(t) at r0. Obviously, such a definition is meaningful only if the realizations are everywhere differentiable with probability one.A necessary condition for this is thatas -> 0, for some finite positive constant C. See for example Cramer [5].Many processes considered in the literature do not satisfy (1.2).See, for example Slepian [10].In this paper we introduce a new but natural definition of an upcrossing.For any e>0, we say that an "e-upcrossing" of the level x occurs at r0 if X(t0) = x, and X(t) 1 such that lim sup |log t\B(\-r(t)) < oo.