Jonction maximale en distribution dans le cas markovien

Résumé Une représentation des sousmartingales retournées est utilisée pour établir que le moment de jonction maximale en distribution peut être choisi comme un temps d'arrêt randomisé pour les chaînes de Markov (non homogènes) de mêmes probabilités et de distributions initiales différentes. En outre nous illustrons, à partir d'un exemple, la non-unicité de la jonction maximale en distribution et ceci malgré des conditions supplémentaires.

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Summary A representation of reversed submartingales is used to establish that the epoch of a maximal distributional junction can be chosen as a randomized stopping time for the Markov chains (inhomogeneous) with different initial distributions and the same transition probabilities. Also, we give an example to show that the maximal distributional junction is not unique even if additional conditions are imposed.

     

Grandes déviations du temps local du mouvement brownien fractionnaire

The purpose of this paper is to prove a large deviation principle for a local time of fractional Brownian motion B^H for all H∈(0,1). To cite this article: E.H. Lakhel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 797–801.     

Approximation du temps local des surfaces gaussiennes

Summary LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d2),F its spectral measure, (x denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formX (t)=( *X)(t) where stands for an approximation of unity (as tends to zero) satisfying certain regularity conditions.The aim of this paper is to recover the local time ofX at a given levelu, as a limit of appropriate normalizations of the geometric measure of theu-level set of the regular approximating processesX . A part of the difficulties comes from the fact that the geometric behavior of the covariance of the Gaussian processX can be a complex one as approaches O.The results are onL ²-convergence and include bounds for the speed of convergence.L ^presults may be obtained in similar ways, but almost sure convergence or simultaneous convergence for the various values ofu do not seem to follow from our methods. In Sect. 3 we have included examples showing a diversity of geometric behaviors, especially in what concerns the dependence on the thickness of the set in which the covariance of the original processX is irregular.Some technical results of analytic nature are included as appendices in Sect. 4.     

Probability density function of the local score position

We calculate the probability density function of the local score position on complete excursions of a reflected Brownian motion. We use the trajectorial decomposition of the standard Brownian bridge to derive two different expressions of the density: the first one is based on a series and an integral while the second one is free off the series.     

Sur les transformées de Riesz dans le cas du Laplacien avec drift

We prove estimates for Riesz transforms with drift.

     

Distribution statistique de l'ordre d'un element du groupe symetrique

Sans résumé     

Un analogue local du théorème de Harnack

Sans résumé     

Extension du filtre de Chandrasekhar au cas des modèles espace d'état périodiques

This Note extends the Chandrasekhar-type recursions due to Morf, Sidhu, and Kailath (1974) to the case of periodic time-varying state-space models. We show that the S-lagged increments of the one-step prediction error covariance satisfy certain recursions from which we derive some algorithms for linear least squares estimation for periodic state-space models. The proposed recursions have potential computational advantages over the Kalman Filter and, in particular, the periodic Riccati difference equation. To cite this article: A. Aknouche, F. Hamdi, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Étude du cas rationnel de la théorie des formes linéaires de logarithmes

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v₀ be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u∈Lie(G(Cv₀)) a logarithm of a point p∈G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)k_⊗Cv₀. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height loga of p, removing a polynomial term in logloga. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.     

Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

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Support partill du CICYT, N^o PB86-0238     

Hyperbolicité du complémentaire d'une courbe dans P2 : le cas de deux composantes

We prove that the complement of a very generic complex curve with two components in $\text{P}{}^2$ of degrees d₁?d₂ is hyperbolic in the sense of Kobayashi in the following cases: d₁?5; d₁=4 and d₂?7; d₁=d₂=4; d₁=3 and d₂?9; d₁=2 and d₂?12. We consider logarithmic jets developped by Dethloff and Lu (Osaka J. Math. 38 (2001) 185–237), who generalized to the logarithmic situation Demailly's jet bundles (in: Proc. Sympos. Pure Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 285–360), and used by El Goul to obtain results about the hyperbolicity of the complement of a very generic curve in $\text{P}{}^2$ in the case of a single component. To cite this article: E. Rousseau, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Approximation du temps local des processus gaussiens stationnaires par régularisation des trajectoires

Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L ² quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.

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Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕ_ɛ(u=ɛ⁻¹ ϕ(u/ɛ) be an approximate identity. Setting X _ɛ(t)=X*ϕ_ɛ(t) and letting N _ɛ(T) be the number of zeros of X _ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N _ɛ(T) converges in L ² as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.

     

Distribution of the supremum of increments of Brownian local time

The joint distribution of the variables and where t(t,x) is Brownian local time, is determined uniquely by the Laplace transform. The computation of this transform constitutes the basic content of this paper. The obtained expression is used for the derivation of the exact modulus of continuity of the process t(t, x) with respect to the variable x:.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 6–24, 1985.     

Congruences entre séries d'Eisenstein,dans le cas supersingulier

Sans résuméNous remercions ici J-P. Serre ainsi que J. Coates, R. Elkik, J. Fresnel, Ph. Satgé et J. Vélu qui ont chacun contribué à l'élaboration de ce texte.     

Les extensions quadratiques du corps des rationnels,ou du corps de Causs,ou du corps des racines cubiques de l'unité de calibres 1

Résumé Nous transposons certains des résultats sur le problème du nombre de classes 1 établis dans le cas des extensions quadratiques réelles de ℚ au cas des extensions quadratiques de ℚ(i) et ℚ(j): nous caractérisons (en terme de contrainte sur leurs discriminants relatifs) les extensions quadratiques de ℚ(i) et ℚ(j) dont la classe principale est de calibre 1, puis déterminons, sous l'assomption d'une forme convenable de l'hypothèse de Riemann, ces extensions quadratiques de calibre 1, i.e. ces extensions à classe principale de calibre 1 qui sont principales. Le cas des extensions quadratiques de ℚ(j) est particulièrement satisfaisant: après avoir amendé dans ce cadre les valeurs des bornes de Minkowski jusqu'à ce jour connues, nous disposerons d'une caractérisation de principalité des extensions quadratiques de ℚ(j) qui se spécialisera en une condition nécessaire et suffisante pour qu'une telle extension soit de calibre 1. La motivation de cette détermination repose sur le résultat de R Paysant-Le Roux suivant lequel il n'existe qu'un nombre fini de corps de nombres de degré et de calibre donnés. Notons que nos résultats s'étendent sans difficulté au cas des extensions quadratiques d'une des neuf corps quadratiques imaginaires principaux, et ce d'autant plus aisément que les deux cas que nous traitons ici sont ceux qui sont source du plus de difficulté du fait qu'ils correspondent aux corps quadratiques imaginaires pour lesquels le groupe des racines de l'unité n'est pas réduit à {−1,+1}.

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