Searcharxiv⌕ Search

arXiv subjects

Jean-Marc Azais

Publications and source records attributed to Jean-Marc Azais.

6 recordsLinked to original sources
arXiv2020

Mean number and correlation function of critical points of isotropic Gaussian fields and some results on GOE random matrices

Let $\mathcal{X}= \{X(t) : t \in \mathbb{R}^N \} $ be an isotropic Gaussian random field with real values.In a first part we study the mean number of critical points of $\mathcal{X}$ with index $k$ using random matrices tools.We obtain an exact expression for the probability density of the $k$th eigenvalue 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 proved. The correlation function between maxima (or minima) depends on the dimension of the ambient space.

math.PR↗
arXiv2018

Conditions for the finiteness of the moments of the volume of level sets

Let X(t) be a Gaussian random field R d $\rightarrow$ R. Using the notion of (d -- 1)-integral geometric measures, we establish a relation between (a) the volume of the level set (b) the number of crossings of the restriction of the random field to a line. Using this relation we prove the equivalence between the finiteness of the expectation and the finiteness of the second spectral moment matrix. Sufficient conditions for finiteness of higher moments are also established.

math.PR↗
arXiv2014

Spike detection from inaccurate samplings

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments...) using an $\ell_{1}$-regularization procedure. In particular, we provide an explicit quantitative localization of the spikes.

math.ST↗
arXiv2013

The record method for two and three dimensional parameters random fields

Let $S$ be a regular set of $\R^d$ and $X : S\rightarrow \R$ be Gaussian field with regular paths. In order to give bound to the tail of the distribution of the maximum, we use the record method of Mercadier. We present some new form in dimension 2 and extend it to dimension 3 using the result of the expectation of the absolute value of quadratic forms by Li and Wei. Comparison with other methods is conducted.

math.PR↗
arXiv2005

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.

math.PR↗