Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Un théorème de représentation pour les martingales discontinues

We consider a fixed integer-valued random measure (also called a random point process). We represent any local martingale as the sum of a stochastic integral with respect to this random measure, and of a local martingale which does not jump on the support of the random measure.     

Un théorème de dualité

Sans résumé Herrn H. Hopf zum sechzigsten Geburtstag gewidmet     

Un théorème sur les séries de Dirichlet

We considerk Dirichlet series a _j (n)n ^–s (1jk),k2. We suppose that for eachj the series a _j (n)n ^–s converges fors=s _j =j+it _j , and that Max j<1/(k–1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line es = (₁+...+ _k )/k+1–1/k and we estimate the rate of convergence. The number 1–1/k cannot be replaced by a smaller one.     

Un théorème de décomposition pour les polaires génériques d'une courbe plane

In this Note we state a decomposition theorem into bunches of branches (i.e. analytically irreducible germs) for the generic polar curves of a reduced germ of a plane analytic curve, with equation ƒ(x, y) = 0. They are the curves with equation ∂f(x, y)/∂yτ∂f(x, y)/∂x = 0 with generic τ. All the branches of the same bunch have the same contact with each branch of C. A number of the first terms of the Puiseux expansion of each branch of the polar is therefore independent of τ; this number depends only on the bunch to which the branch belongs. This generalizes results of H.J.S. Smith [8], M. Merle [7] (where C is a branch), E. Casas [1], F. Delgado [2]. We show by an example that it is also optimal. We have shown elsewhere that it implies the results of Lê-Michel-Weber [6].     

Le théorème d'approximation pour les valuations deKrull

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 <.

     

Un théorème de Torelli infinitésimal à coefficients

We generalize an infinitesimal Torelli theorem for hypersurfaces of high degree of M. Green (1985) [1], for twisted coefficients.     

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let be a quotient of X by a torsion-free discrete subgroup of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual X _c , the locally symmetric structure of M can be considered as a special kind of a -structure on M, a maximal atlas of X _c -valued charts with locally constant transition maps in the complexified group . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung. Received: September 23, 1999     

Théorème de Poincaré–Alexander pour les domaines modèles

The Poincaré–Alexander Theorem states that holomorphic mappings defined on an open subset of the unit ball of ${\mathbb{C}}^n$ may, under certain conditions, be extended to a biholomorphism of the unit ball. In a complex manifold, every strongly pseudoconvex homogeneous domain is biholomorphic to the unit ball. In an almost complex manifold, the unit ball is not the only strongly pseudoconvex homogeneous domain. A strongly pseudoconvex homogeneous domain is biholomorphic to a model domain. The aim of this paper is to extend this theorem to model domains.     

Un théorème de Harris et Segal pour un anneau d'entiers algébriques exceptionnel

Let K be a number field, O_K the ring of the integers of K, ℓ a prime integer and Z_(ℓ) the localisation of Z at ℓ. Harris and Segal [4] proved that there exists infinitely many primes p of O_K such that the natural morphism K_i(O_K) ⊗ Z_(ℓ) → K_i(O_K /p) ⊗ Z_(ℓ) in algebraic K-theory is split surjective for i > 0, except if ℓ = 1 and K is exceptional. In this Note, we prove that the Harris-Segal theorem is still true for ℓ = 2 in the exceptional case, if we replace algebraic K-theory by orthogonal K-theory defined by Karoubi [5]. Thanks to [3], we can then determine a direct summand of the 2-torsion of KO_n(O_K).     

Le problème des diviseurs de zéro pour les groupes de Lie nilpotents

Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist $0\ne \alpha \in C_c(G)$ and $0\ne \xi \in L^2(G)$ such that $\alpha 1\xi =0$, contrary to what happens for the group ${\mathbb{R}}^n$. Moreover, the set of zero divisors is a total subset of $L^2(G)$. This result is first proven for the Heisenberg group $H_n$ where it is based on the existence of non-trivial Schwartz functions f satisfying $f1(X_k+\mathrm{i}{Y}_k)=0$ for $1?k?n$. To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).