On the distribution of the maximum of a gaussian field with d parameters

Let I be a compact d-dimensional manifold, let X:I\to R be a Gaussian process with regular paths and let F_I(u), u\in R, be the probability distribution function of sup_{t\in I}X(t). We prove that under certain regularity and nondegeneracy conditions, F_I is a C^1-function and satisfies a certain implicit equation that permits to give bounds for its values and to compute its asymptotic behavior as u\to +\infty. This is a partial extension of previous results by the authors in the case d=1. Our methods use strongly the so-called Rice formulae for the moments of the number of roots of an equation of the form Z(t)=x, where Z:I\to R^d is a random field and x is a fixed point in R^d. We also give proofs for this kind of formulae, which have their own interest beyond the present application.Comment: Published at http://dx.doi.org/10.1214/105051604000000602 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mean number and correlation function of critical points of isotropic Gaussian fields

Let X = {X(t) : t ∈ R N } be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce some exact expressions for the mean number of critical points with a given index. In a second part we study attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space

Computing the Distribution of the Maximum of a Gaussian Process

This paper deals with the problem of obtaining methods to compute the distribution of the maximum of a one-parameter stochastic process on a fixed interval, mainly in the Gaussian case. The main point is the relationship between the values of the maximum and crossings of the paths, via the socalled Rice's formulae for the factorial moments of crossings. In certai

Analyse de variance non orthogonale avec GLM/SAS

SIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 22522, issue : a.1992 n.2 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Modeles de proximite inadequats Etude de validite par simulation

Available at INIST (FR), Document Supply Service, under shelf-number : 22522, issue : a.1992 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc

Proton Flux Anisotropy in the Atmosphere: Experiment and Modeling

International audienceWe used two detectors differently tilted during stratospheric balloon flights and we proved the proton flux anisotropy

Experimental Characterization of Atmospheric Radiation Environment with Stratospheric Balloon

International audienceWe report a stratospheric flight with a CNES balloon for which we developed a silicon detector in order to obtain data on the atmospheric radiation environment. The number of detected protons is shown to be directly correlated with the altitude. Simulations with the MC-ORACLE code, which uses pre-calculated fluxes with QARM, are in good agreement with our experimental results

Asymptotic expansions for the distribution of the maximum of Gaussian random fields

SIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : 22522, issue : a.2001 n.2 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc