Processus hilbertien associé à la convolée de deux mesures aléatoires

For two random measures (in the [1] sense) the tensor and convolution products are defined and a Fubini type formula is given. The results are applied to Hilbertian stationary continuous random functions defined on an Abelian locally compact group.     

Convergence de mesures spectrales aléatoires et applications à des principes d'invariance

Nous présentons un cadre unificateur pour un certain nombre de résultats connus de convergence en loi pour des processus admettant une représentation linéaire par rapport à un bruit blanc faible pour lequel il existe un principe dinvariance faible. Nous montrons que ce principe dinvariance est équivalent à la convergence dune suite de mesures aléatoires spectrales. Ceci permet daffaiblir les hypothèses usuelles et donc dobtenir des résultats nouveaux de convergence en loi, notamment pour des processus à longue portée.     

Sur les séries aléatoires de Dirichlet

The Borel-Cantelli lemma, a method of J.-P. Kahane, and an extension of a lemma of Paley-Zygmund are applied to study random Taylor and Dirichlet series.     

Principes de grandes déviations pour l’estimateur à noyau de la densité de probabilité

We establish pointwise and uniform large deviations principles for nonparametric kernel density estimators. The estimates are based on sequences of independent and identically distributed real random variables.     

Homologies associées aux variétés de Poisson

In this article, we obtain results on the cyclic homology of Poisson manifolds as defined by Kassel. In particular, we compute the cyclic homology of any symplectic variety. We also show that there exists a family of new operators of the type of the S operator of Connes, defined on the cyclic homology of a Poisson manifold, and describe the associated exact triangles associated to these new operators.

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Received February 23, 1998 / Published online July 20, 2000     

Sous-laplaciens et densités centrées sur les groupes de Lie à croissance polynomiale du volume

We prove a Harnack inequality on connected Lie groups of polynomial volume growth. We use this inequality to study the large time behavior of the heat kernels associated to centered sub-Laplacians. Thus, we obtain Gaussian estimates and estimates of the type Berry—Esseen. We also obtain similar results for the convolution powers of centered densities.     

Classification de certains feuilletages associés à un cusp

We study singularities of foliations given by R. Moussu closely related to a conjecture of R. Thom. We give their analytic classification and prove their topological rigidity.

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Notations , ,F 0 - H 0, III - H ₁,H _p I - H-cusp,H _r -cusp I-3 - H _p I-3, III-2 - -cusp II-1 - _r -cusp III-1     

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.     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

Cascades aléatoires et équations de Navier-Stokes

This work concerns the incompressible Navier-Stokes equation in. The non linear integral equation satisfied bx the Fourier transform of the Laplacian of the velocity field can be interpreted in terms of a branching Markov process and of a composition rule defined along the associated tree. We derive from this representation new classes where global existence and uniqueness can be proven.     

Sur deux congruences associées à une congruence W

Sans résuméEn l'honneur de M. E. Kruppa     

Inégalités d'incertitude associées à des fonctions homogènes

The uncertainty principle states that a nonzero function and its Fourier transform cannot be both sharply localized. It is well known that the support and the spectrum of a function cannot both have finite measure. In Shubin et al. [Geom. Funct. Anal. 8 (1998) 932–964], it is shown that they cannot be contained in ?-thin sets E and $\stackrel{?}{E}$. We give here other examples of sets E and $\stackrel{?}{E}$ having this property. The thinness of the sets is expressed in terms of a dyadic decomposition of the space, which is related to the functions ${|x_1|}^{{\alpha }_1}?{|x_d|}^{{\alpha }_d}$ on ${\mathbb{R}}^d$. These sets are not ?-thin in general. We prove that the pairs of sets we consider are strongly annihilating in the sense of Havin and Jöricke. To cite this article: B. Demange, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimation de la densité de probabilité améliorée par pre-testing

We consider the nonparametric problem of multidimensional probability density estimate. Using concept of minimax risk with random normalizing factor introduced by Lepski [Math. Methods Statist. 8 (1999) 441–486], by considering an independence hypothesis, we build an estimator which can be adaptive and whose accuracy, depending on the observation, is better than the minimax estimate, $n^{?\frac{\beta }{2\beta +d}}$, with prescribed confidence level. To cite this article: A.F. Yode, C. R. Acad. Sci. Paris, Ser. I 340 (2005).