Estimation de la fonction de régression pour variable explicative et réponse fonctionnelles dépendantes

We consider a kernel estimate of the regression when the response variable is in a Banach space and the explanatory variable takes its values in a semi-metric space. Our main result states the almost complete convergence (with rate) of the constructed estimate when the sample considered is a β-mixing sequence.     

Estimation non paramétrique de la densité d'un processus stationnaire mélangeant

In this Note, we consider the density estimate of Loader [2] for a stationary strongly mixing process. We establish, under some mild assumptions, the asymptotic properties of this estimate.     

Un algorithme de classification automatique non paramétrique

We present a new clustering algorithm based on the search for a stable partition with respect to a non parametric discrimination rule. A formal link between this method and the Hopfield connectionist model is shown, allowing for the definition of an energy function whose decrease at each iteration guarantees the stationarity of the sequence of partitions obtained. A link between this approach and the principle of classification likelihood maximisation is also studied.     

Estimation non paramétrique du contour du support d'un processus ponctuel de Poisson

We use one method type kernel estimation with random window for estimate the edge of support of a Poisson point process. The estimator obtained converges uniformly in probability, almost surely, almost completely and converges weakly.     

Estimation locale linéaire de la densité conditionnelle pour des données fonctionnelles

In this Note, we introduce the local linear estimation of the conditional density of a scalar response variable given a random variable taking values in a semi-metric space. Under some general conditions, we establish the pointwise and uniform almost complete convergences with rates of this estimator. Moreover, as an application, we use the obtained results to derive some asymptotic properties for the local linear estimator of the conditional mode.     

Propriétés asymptotiques d'estimateurs convergents des quantiles conditionnels

We state sufficient conditions for asymptotic normality of convergent estimates of the conditional quantiles, irrespective of data dependence, and give an application to α-mixing stationary processes, under optimal conditions. As an application, we use asymptotic normality to construct confidence bands for predictors based on nonparametric estimates of the conditional median.     

Topologie locale des méthodes de Newton cubiques: plan paramétrique

Cubic Newton 's methods are rational maps having three distinct super-attracting fixed points and a single free critical point. They form, up to conjugation, a family N_λ parametrized by Λ = ℂ\{0,±3/2}, and we denote by ℋ₀ the set of λ for which the free critical point of N_λ is in the immediate basin of one of the super-attracting fixed points. In this Note, we show that the boundary of each connected component of ℋ₀ is a Jordan curve. For this, we determine in Λ regions on which the dynamics of N_λ can be described by a fixed combinatorial model.     

Sur l'intégration des fonctions presque-périodiques des deux variables indépendantes

Sans résumé     

Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes

Résumé Nous étendons la méthode de démonstration du théorème de Berry-Esseen proposée par Bergström aux suites de variables aléatoires faiblement dépendantes. En particulier, nous montrons que, pour les suites stationnaires de variables aléatoires réelles bornées, la vitesse de convergence dans le théorème limite central en distance de Lévy est de l'ordre den ^–1/2 dès que la suite ( _p)p>0 des coefficients de mélange uniforme satisfait la condition _p>0 p _p <

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About the Berry-Esseen Theorem for weakly dependent sequences

We extend the method of Bergström for the rates of convergence in the central limit theorem to weakly dependent sequences. In particular, we prove that, for stationary and uniformly mixing sequences of real-valued and bounded random variables, the rate of convergence in the central limit theorem is of the order ofn ^–1/2 as soon as the sequence ( _p)p>0 of uniform mixing coefficients satisfies _p>0 p _p <.

     

Modèle topologique pour des applications de Hénon complexes

In order to describe the dynamics of the complex Hénon map $H_{a\text{,}c}:\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \end{array}\right)?\left(\begin{array}{c}\hfill P_c(x)?ay\hfill \\ \hfill x\hfill \end{array}\right)$, where $P_c:z?z^2+c$ has an attractive fixed point, we build a topological model $(g\text{,}Y)$. In this model Y is the complement in ${\mathbb{R}}^4$ of a cone over a solenoid lying in the unit 3-sphere, and $g\text{:}Y\to Y$ is a map given in spherical coordinates by $g(r\text{,}\theta )=(r^2\text{,}\sigma (\theta ))$, where σ is a solenoidal map of degree two. Then we prove the existence of a constant $\mathrm{\epsilon }>0$ such that any Hénon map $H_{a\text{,}c}$ with $0<|a|<\mathrm{\epsilon }$ is conjugate to our model $(g\text{,}Y)$. To cite this article: S. Bonnot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Analyse numérique de la méthode des variables adiabatiques pour l'approximation de l'hamiltonien nucléaire

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed, these eigenfunctions depend on 3n variables, where n stands for the number of particles (electrons or protons) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many interesting ideas. Among those stands the adiabatic variable method; we present in this Note a mathematical analysis of this approximation and propose, in particular, an a posteriori estimate that might allow for verifying the adiabaticity hypothesis that is done on some variables.