Level crossings and other level functionals of stationary Gaussian processes

Abstract

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some exten-sions. The main results [(factorial) moments, representation into theWiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal com-parison method, Rice method, Stein-Chen method, a general m-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupa-tion functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.