Mesures majorantes et loi du logarithme itéré pour les variables aléatoires sous-gaussiennes

It is shown that for the elements X of a large class of subgaussian random variables with values in (C(S), |·|_∞) the existence of a probability measure λ on S such that:

$\text{lim}\text{ε←0}\text{sup}\text{s∈S}\text{∫}\text{0}\text{ε}\text{ψ(}\text{1}\text{κ}{}^2\text{{t:(E(X)(s)?X(t))}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{<u})du=0}$

(where ψ denotes the inverse function of x → exp x²L₂x) is sufficient to imply that X satisfies the law of the iterated logarithm.     

Loi du logarithme itéré dans C S) et fonction caractéristique empirique

Sommaire De nombreuses publications ont été consacrées ces dernières années à l'étude des propriétés limites des variables aléatoires lipschitziennes. Si le théorème de la limite centrale a trouvé dans les travaux de N.C. Jain et M.B. Marcus [9] et B. Heinkel [7] des conditions suffisantes en un sens meilleures possibles, la loi du logarithme itéré, bien qu'apparaisant comme corollaire immédiat de la propriété de limite centrale, était toujours à la recherche de conditions spécifiques. Nous présentons, dans notre première partie, une loi du logarithme itéré pour les variables aléatoires lipschitziennes sous des hypothèses qui lui sont propres. Dans la seconde, nous appliquons les techniques du cas lipschitzien à l'étude de la fonction caractéristique empirique en vue de préciser, dans leurs parties loi du logarithme itéré, les récents résultats de M.B. Marcus [15] et S. Csörg [3] concernant les propriétés limites de la fonction caractéristique empirique.     

Inégalité du discriminant pour les pinceaux elliptiques à réductions quelconques

In this article the degree of the discriminant of an elliptic pencil on a projective curve is upper-bounded by using the degree of its conductor and the genus of the base curve. This is done in the most general case, extending a method and a result of Szpiro (1981 and 1990a) and a result of Hindry and Silvermann. The difficult part, dealing with characteristic 2 and 3 and additive reductions, need the introduction of a new object - which we called 'conducteur efficace' - defined by using differentials and interestingly comparable to the usual conductor. This article ends with a few results in the arithmetical case - case corresponding to an inequality conjectured by the second author in 1978: (1) the proof of this inequality in the potentially good reduction cases; (2) the passage from the semi-stable reduction to the general case for a strong inequality.     

Un résultat de stabilité pour les solutions faibles des équations des fluides tournants

We study the limit of the periodic, incompressible, rotating fluid equations, as the Coriolis force goes to infinity: in the case of well-prepared initial data in L², the weak solutions converge to the solution of a two-dimensional, incompressible Navier-Stokes equation. We also prove that the rotating fluid equations are globally well-posed under an appropriate assumption on the oscillating part of the initial data.     

Régularité de la loi du maximum de processus gaussiens réguliers

Let X be a real-valued Gaussian process with a one-dimensional parameter. If the sample paths of X are of class C^2k, k integer and if the marginal distributions of X and its derivatives have no degeneracy, then the distribution function of the maximum on a bounded interval is of class C^k. We give a recurrence method allowing to express the derivatives by means of certain implicit formulae.     

Un test d'auto-similarité pour les processus gaussiens à accroissements stationnaires

In this Note, we present a method for testing self-similarity of discretized observations of a Gaussian process with stationary increments. The test is based on the estimation of a distance between the process and a set of processes containing all the fractional Brownian motions. This distance is constructed from two estimations of multiscale generalized quadratic variations expectations. The second estimation requires to estimate by regression the self-similarity index H. Both these estimators of H present good robustness and computing time properties compared with maximum likelihood approach, with nearly similar convergence rate.     

Un résultat de densité pour les équations de Maxwell

We prove the density of regular fielclx in the space of square-integrahle fields with square-integrable curls and square-integrable tangential traces on the boundary. This space is involved in one of the variational formulations of the electmmagnetism equations.     

Eléments finis nodaux pour les équations de Maxwell

An original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: $\mathbf{E}={\mathbf{E}}^R+\lambda {\mathbf{x}}_P$, where ${\mathbf{E}}^R\in H^1{\left(\omega \right)}^2$, λ depends on the data and domain and ${\mathbf{x}}_P$ is known explicitly. The same splitting can be used for the magnetic field. No cut-off function is needed and improved error estimates are derived. To cite this article: E. Jamelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Dualité tannakienne pour les quasi-groupoïdes quantiques

This paper is the sequel of a previous one [2] where we extended the Tannaka-Krein duality results to the non-commutative situation, i.e. to ‘quantum groupoids’. Here we extend those results to the quasi-monoidal situation, corresponding to ‘quasi-quantum groupoids’ as defined in [3] (‘quasi-’ stands for quasi-associativity a la Drinfeld). More precisely, let B be a commutative algebra over a field k. Given a tensor autonomous category τ,. we define the notion of a quasi-fibre functor ω:τ-proj B (here, ‘quasi-’ means without compatibility to associativity constraints). On the other hand, we define the notion of a transitive quasi-quantum groupoid over B. We then show that the category of tensor autonomous categories equipped with a quasi-fibre functor (with suitable morphisms), is equivalent to the category of transitive quasi-quantum groupoids (5.4.2)

Moreover, we classify quasi-fibre functors for a semisimple tensor autonomous category (6.1.2), and give a few examples : a family of quantum groups having the same tensor category of representations as Sl²(C), but with non-isornorphic underlying coalgebras, constructed by means of an R-matrix introduced by Gurevich ([9]) in a manner suggested to the author by Lyubashenko (6.2.1 and 6.2.2), and quasi-quantum groups which cannot be obtained from quantum groups by a Drinfeld twist (6.2.1)     

Cohomologie des formes différentielles régulières pour les courbes affines

Let C be an affine curve, and denote by H¹(C) its first troncated De Rham cohomology group, i.e. the quotient of regular differential 1-forms on C by exact 1-forms. First we introduce a nonnegative invariant μ′(C,x) that measures the complexity of the singularity of C at the point x, and we establish the following formula:

$\text{dim}\text{H}{}^1\text{(C)=}\text{dim}\text{H}{}_1\text{(C)+}\text{∑}\text{x∈C}\text{μ′(C,x),}$

where H₁(C) is the first singular homology group of C with complex coefficients. Second we consider a family of curves given by the fibres of a morphism $\text{f}\text{:X→}\text{C}$, where X is an affine reduced surface. We analyse the behaviour of the function y?dimH¹(f^?1(y)). More precisely we show that it is constant over a Zariski open set, and that it is lower semi-continuous is general. To cite this article: P. Bonnet, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Une généralisation du problème des itérés

For given systems of differential operators of constant coefficients we give algebraic conditions to compare their spaces of Gevrey vectors.     

Normalité asymptotique de l'estimateur par polynômes locaux de la densité et ses dérivées pour des processus réels et mélangeants

We establish the convergence in probability and the asymptotic normality of the kernel polynomial estimators of the density function and its derivatives when the observed process is strictly stationary and strong mixing.     

Sur les valeurs moyennes des fonctions réelles définies pour toutes les valeurs de la variable

Sans résumé     

Quelques nouvelles identités de fluctuation pour les processus de Lévy

Let τ and H be the ladder time and ladder height processes of a Lévy process X. We give an identity in law between (τ,H) and (X,H^*), H^* being the right continuous inverse of the process H. The later allows us to get a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (X_t- inf_s≤ X_s >- 0). In the stable case, some explicit calculations are provided.