Probability Surveys

Level crossings and other level functionals of stationary Gaussian processes

Marie F. Kratz

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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.

Article information

Probab. Surveys, Volume 3 (2006), 230-288.

First available in Project Euclid: 19 December 2006

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G10: Stationary processes 60G12: General second-order processes 60G60: Random fields 60G70: Extreme value theory; extremal processes 60F05: Central limit and other weak theorems

(up) crossings (non) central limit theorems Gaussian processes/fields Hermite polynomials level curve level functionals local time (factorial) moments normal comparison method number of maxima Poisson convergence rate of convergence Rice method sojourn Wiener chaos


Kratz, Marie F. Level crossings and other level functionals of stationary Gaussian processes. Probab. Surveys 3 (2006), 230--288. doi:10.1214/154957806000000087.

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