Level crossings and other level functionals of stationary Gaussian processes

Details Level crossings and other level functionals of stationary Gaussian processes Level crossings and other level functionals of stationary Gaussian processes

2006-12-20

arxiv.org/abs/math/0612577v1

Probability Surveys 2006, Vol. 3, 230-288

Mathematics

Probability

Published at http://dx.doi.org/10.1214/154957806000000087 in the Probability Surveys (http://www.i-journals.org/ps/) by the In

Scientific paper

10.1214/154957806000000087

This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. 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 comparison 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 occupation 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.

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