On statistics at level crossings by a stationary process

Abstract

The main purpose of this paper is to indicate formal theorems and conditions for the extensions of Rice’s Formula for intensities of crossings of a fixed level u by a stationary process X(t) in order to provide the conditional (Palm) distribution of an associated process Y(t) at the level crossing instants. Sections 1 and 2 provide brief background material on Rice’s Formula and Palm distributions, while relevant historical comments and basic results are given in Section 3. These are stated with some indications of proof in Section 4 and calculation of the Palm distributions shown in Section 5. Two cases of importance in applications are considered: (a) where (roughly) X(t), Y(t) and the derivative X′(t) have a joint density and (b) where Y(t)=X′(t). The resulting Palm distributions are each absolutely continuous with readily calculable densities. These are evaluated for Gaussian processes in Section 6 and applications to motivating 2 structural safety questions indicated in Section 7.