On crossings of arbitrary curves by certain Gaussian processes

Abstract

under conditions which are very close to the necessary ones. Here N(T) is the number of crossings of the level u in (0, T) by the stationary Gaussian process x(t), with covariance function r(-r). The symbol E denotes expectation. The treatment of this problem given by Bulinskaya is essentially a rigorization of the method used by Grenander and Rosenblatt [4], which in turn extends an argument due to Kac [7]. For some applications, however, it is important to consider crossings of a curve, rather than a fixed level, by such a process, and to consider also the same problem for certain Gaussian, but nonstationary processes. In ?2 we shall obtain the formula corresponding to (1) for the case where { x(t) } is a stationary Gaussian process and u = u(t) is an arbitrary (differentiable) curve, instead of a fixed level. In ?4, the same problem will be considered for a nonstationary process z(t) = f,x(s)ds where { x(t) } is, as before, a stationary Gaussian process. The discussion of this z(t)-process has application to the study of the probabilistic behaviour of certain physical systems. It would be possible to state a result corresponding to (1) for curve crossings by a member of a wide class of (nonstationary) Gaussian processes. However, this could be stated in very general terms only, and moreover it is obvious from the derivation for the z(t)-case how such a general result would be formulated.