Level crossings and other level functionals of stationary Gaussian processes

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.

Level curves crossings and applications for Gaussian models

“…Hermite polynomial expansion or Multiple Wiener-Itô Integrals (MWI) may be a powerful tool to represent and to study nonlinear functionals of stationary Gaussian processes (see Kratz andLeón 1997 or Slud 1994). One advantage of using methods based on Hermite polynomial expansions rather than on MWI, is to obtain asymptotic behavior of such functionals in an easier way and under explicit conditions.…”

“…In this work, we also use generalizations of the well known Rice formula (see Rice 1944) and apply the general method we settled in earlier works (see Kratz and León 1997, 2001, or Kratz 2006 to obtain representations into the Itô-Wiener Chaos and asymptotic behaviors for level functionals of stationary Gaussian processes, having in mind applications to physics and sea modelling.…”

On level crossings for a general class of piecewise-deterministic Markov processes

“…[18], [25], and the references therein. To the best of the authors' knowledge, the present paper is the first to establish such a limit theorem for jump processes.…”

“…Therefore, the quantities 18) do not depend on v. Now we assume that µ(∞) < 0. Then crossing level b can only occur due to a jump, and since to get from level x > b down to level u will require a continuous downcrossing of b, we obtain…”

On the Detection of Periodic Components in Observational and Experimental Data

“…Recently, a significant progress was attained in this field by the author [2]- [5]. In these papers, the approximations to the distribution function of the maximum are obtained using the so-called Rice method, which is based on the analysis of level up-crossings [1], [15]. We present these results below.…”