arXiv subjects

Jean-Marc Azaïs

Publications and source records attributed to Jean-Marc Azaïs.

22 records · Page 2Linked to original sources

arXiv2015

CLT for the zeros of Classical Random Trigonometric Polynomials

We prove a Central Limit Theorem for the number of zeros of random trigonometric polynomials of the form $K^{-1/2}\sum_{n=1}^{K} a_n\cos(nt)$, being $(a_n)_n$ independent standard Gaussian random variables. In particular, we prove the conjecture by Farahmand, Granville & Wigman that the variance is equivalent to $V^2K$, $0<V^2<\infty$, as $K\to\infty$. % The case of stationary trigonometric polynomials was studied by Granville & Wigman and by Aza\"\is & León. Our approach is based on the Hermite/Wiener-Chaos decomposition for square-integrable functionals of a Gaussian process and on Rice Formula for zero counting.

math.PR↗

arXiv2011

The tail of the maximum of smooth Gaussian fields on fractal sets

We study the probability distribution of the maximum $M_S $ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $¶(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field.

math.PR↗

arXiv2011

Rice formulae and Gaussian waves

We use Rice formulae in order to compute the moments of some level functionals which are linked to problems in oceanography and optics: the number of specular points in one and two dimensions, the distribution of the normal angle of level curves and the number of dislocations in random wavefronts. We compute expectations and, in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some cases, we are able to use these computations to obtain a central limit theorem.

math.PR↗

arXiv2009

Some applications of Rice formulas to waves

We use Rice's formulas in order to compute the moments of some level functionals which are linked to problems in oceanography and optics. For instance, we consider the number of specular points in one or two dimensions, the number of twinkles, the distribution of normal angle of level curves and the number or the length of dislocations in random wavefronts. We compute expectations and in some cases, also second moments of such functionals. Moments of order greater than one are more involved, but one needs them whenever one wants to perform statistical inference on some parameters in the model or to test the model itself. In some case we are able to use these computations to obtain a Central Limit Theorem.

math.PR↗