Level crossings and other level functionals of stationary Gaussian processes

By Marie Kratz. 2006

97 citations recorded by the source corpus.

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 the Wiener 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. AMS 2000 subject classifications: Primary 60G15; secondary 60G10, 60G12, 60G60, 60G70, 60F05. Keywords and phrases: (up) crossings, (non) central limit theorems, Gaussian processes/fields, Hermite polynomials, level curve, level function- als, local time, (factorial) moments, normal comparison method, number of maxima, Poisson convergence, rate of convergence, Rice method, sojourn, Wiener chaos.

Subfields: Economics and Econometrics, Finance

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