Open Access
2006 Level crossings and other level functionals of stationary Gaussian processes
Marie F. Kratz
Probab. Surveys 3: 230-288 (2006). DOI: 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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Marie F. Kratz. "Level crossings and other level functionals of stationary Gaussian processes." Probab. Surveys 3 230 - 288, 2006.


Published: 2006
First available in Project Euclid: 19 December 2006

zbMATH: 1189.60079
MathSciNet: MR2264709
Digital Object Identifier: 10.1214/154957806000000087

Primary: 60G15
Secondary: 60F05 , 60G10 , 60G12 , 60G60 , 60G70

Keywords: (factorial) moments , (non) central limit theorems , (up) crossings , Gaussian processes/fields , Hermite polynomials , level curve , level functionals , Local time , normal comparison method , number of maxima , Poisson convergence , rate of convergence , Rice Method , Sojourn , Wiener Chaos

Rights: Copyright © 2006 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.3 • 2006
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