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 Probability Surveys > Vol. 3 (2006) open journal systems 


Level crossings and other level functionals of stationary Gaussian processes

Marie F. Kratz, ESSEC, MAP5 (Univ. Paris V) and SAMOS-MATISSE (Univ. Paris I)


Abstract
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.

AMS 2000 subject classifications: Primary 60G15; secondary 60G10, 60G12, 60G60, 60G70, 60F05.

Keywords: (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 metho

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Kratz, Marie F., Level crossings and other level functionals of stationary Gaussian processes, Probability Surveys, 3, (2006), 230-288 (electronic).

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Probability Surveys. ISSN: 1549-5787