Multiple Wiener-itô integral expansions for level-crossing-count functionals

Abstract

This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.