On the Distributions of the Times between Events in a Stationary Stream of Events

MANY authors-particularly in the physical and engineering sciences-have considered the problem of obtaining the distribution of the time between successive axiscrossings by a stochastic process. This interest dates back to the pioneering work of S. 0. Rice (1945), who developed a series expression and used a very simple approximation for the case of axis-crossings by a stationary normal process. More generally, one may consider the distribution of the time between an axiscrossing and the nth subsequent axis-crossing, or between, say, an upcrossing of the axis and the nth subsequent downcrossing. Problems of this type have been discussed by Longuet-Higgins (1962), with particular reference to the normal case. In particular he obtains series expressions which are variants and generalizations of that given by Rice (1945, equation 3.4-11). For example, Longuet-Higgins calculates (by somewhat heuristic methods) a series for the probability density for the time between an "arbitrary" upcrossing of zero to the (r+ l)st subsequent upcrossing. (A precise definition of what is meant by such a density is usually not given in the relevant literature. This difficulty, and one method of overcoming it, will be discussed in Section 2.) The series just referred to may be written in the form