Level curves crossings and applications for Gaussian models

Representations into the Itô-Wiener Chaos and asymptotic results such as CLTs are obtained for the curve-crossings number of a stationary Gaussian process according to the form of the curve. Applications in physics and sea modelling follow, with the study of the estimator of the natural frequency of a harmonic oscillator and the study of specular points.

[1]  E. Slud.  MWI Representation of the Number of Curve-Crossings by a Differentiable Gaussian Process, with Applications , 1994 .

[2]  H. Fédérer.  Geometric Measure Theory , 1969 .

[3]  harald Cramer,et al.  Stationary And Related Stochastic Processes , 1967 .

[4]  Reflection and Refraction at a Random Moving Surface. III. Frequency of Twinkling in a Gaussian Surface , 1960 .

[5]  On the second moment of the number of crossings by a stationary Gaussian process , 2006, math/0609682.

[6]  P. Major,et al.  Central limit theorems for non-linear functionals of Gaussian fields , 1983 .

[7]  R. Dobrushin,et al.  Non-central limit theorems for non-linear functional of Gaussian fields , 1979 .

[8]  S. Rice.  Mathematical analysis of random noise , 1944 .

[9]  J. Azaïs,et al.  Level Sets and Extrema of Random Processes and Fields , 2009 .

[10]  Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: Crossings and extremes , 1997 .

[11]  Marie F. Kratz.  Level crossings and other level functionals of stationary Gaussian processes , 2006 .

[12]  M. R. Leadbetter,et al.  Palm distributions of wave characteristics in encountering seas , 2008, 0806.2718.

[13]  M. A. Arcones,et al.  Limit Theorems for Nonlinear Functionals of a Stationary Gaussian Sequence of Vectors , 1994 .

[14]  P. Hänggi,et al.  Oscillatory systems driven by noise: frequency and phase synchronization. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Lam Yeh.  A monotone process maintenance model for a multistate system , 2005 .

[16]  Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization , 1995 .

[17]  M. Longuet-Higgins.  Reflection and Refraction at a Random Moving Surface. I. Pattern and Paths of Specular Points , 1960 .

[18]  E. M. Cabaña.  Affine processes: a test of isotropy based on level sets , 1987 .

[19]  H. Robbins,et al.  The Central Limit Theorem for Dependent Random Variables , 1948 .

[20]  Mario Wschebor,et al.  On the distribution of the maximum of a Gaussian field with d parameters , 2005 .

[21]  Central Limit Theorems for the Number of Maxima and an Estimator of the Second Spectral Moment of a Stationary Gaussian Process, with Application to Hydroscience , 2000 .

[22]  G. Lindgren.  Slepian models for the stochastic shape of individual Lagrange sea waves , 2006, Advances in Applied Probability.

[23]  Mark R. Dennis,et al.  Phase singularities in isotropic random waves , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[24]  Geometrical characteristics of Gaussian sea waves , 2005, Journal of Applied Probability.

[25]  M. Kratz,et al.  Central Limit Theorems for Level Functionals of Stationary Gaussian Processes and Fields , 2001 .

[26]  Igor Rychlik,et al.  Central limit theorem for wave-functionals of Gaussian processes , 1999, Advances in Applied Probability.