Upcrossing probabilities for stationary Gaussian processes 论文

摘要

JAMES PICKANDS IIIO 1. Introduction.Let {X(t), -co<t<co} be a separable, measurable version of a continuous stationary Gaussian stochastic process.In what follows, it will be assumed, without loss of generality, that (1.1) EX(t) = 0, EX2(t) = 1.The probability measure associated with the process, then, is completely determined by the covariance function r(t) = EX(s)X(s+t).By stationarity, of course, r(t) does not depend upon s.Upcrossings and their properties have been studied by a number of authors.They have used the following definition.An upcrossing of the level x is said to have occured at t0 iff X(t0) = x, and X'(t0) > 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) < x, for all t such that t0 -e^t<t0.The advantage of this definition is that it is not necessary to assume that X(t) is differentiable everywhere.It is only necessary to assume that X(t) is continuous everywhere.In order that this be true with probability one, it is sufficient that there exist a > 1 such that lim sup |log t\B(\-r(t)) < oo.