A Bennett concentration inequality and its application to suprema of empirical processesUne inégalité de concentration de type Bennett et son application aux maxima de processus empiriques

We introduce new concentration inequalities for functions on product spaces. They allow to obtain a Bennett type deviation bound for suprema of empirical processes indexed by upper bounded functions. The result is an improvement on Rio's version [6] of Talagrand's inequality [7] for equidistributed variables. To cite this article: O. Bousquet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 495–500.     

Principe de grandes déviations pour l'estimateur de la densité par la méthode des delta-suites

In this Note we obtain pointwise large deviations principle for the delta-sequence method density estimator. A general result is stated for any regular delta-sequence and corollaries with explicit rate functions are derived for delta-sequences, associated to usual estimation methods. The estimation is based upon sequences of independent and identically distributed random variables. To cite this article: N. Berrahou, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Principe de grandes déviations uniforme pour l'estimateur de la densité par la méthode des delta-suites

In this Note, we obtain a large deviations principle for the uniform deviation of the delta-sequence density estimator. A general result is stated for any regular delta-sequence and a discussion of hypotheses for the most usual methods is given. Implications of our results in the study of the inaccuracy rate of these estimates with the comparisons of performances of various methods are given. The estimation is based upon sequences of i.i.d. random variables. To cite this article: N. Berrahou, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Minima locaux,fonctions marginales et hyperplans séparants dans l'optimisation discrète

The goal of this Note is to prove results in optimization of two integer variables which correspond to fundamental results in convex analysis of real variables, viz. that a local minimum of a convex function is global; that the marginal function of a convex function is convex; and that two disjoint convex sets can be separated by a hyperplane. To cite this article: C.O. Kiselman, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Propriétés de l'EDM pour un processus de Poisson d'intensité discontinue

The problem considered is a problem parameter estimation of a 2d-dimensional parameter of a Poisson process. The intensity function of the process is a smooth function with respect to first d variables and is discontinuous function of d other variables. We show the consistency and asymptotic normality of the minimum distance estimator. To cite this article: A.S. Dabye, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On Gronwall-Bellman-Bihari-type integral inequalities in several variables with retardation

Presented are some new nonlinear integral inequalities of the Gronwall-Bellman-Bihari type in n-independent variables with delay which extend recent results of C. C. Yeh and M.-H. Shin [J. Math. Anal. Appl.86, (1982), 157–167], C. C. Yeh [J. Math. Anal. Appl.87, (1982), 311–321], and A. I. Zahariev and D. D. Bainor [J. Math. Anal. Appl.89, (1981), 147–149]. Some applications of the results are included.     

Un algorithme de Schur pour les fonctions de transfert des systèmes surdéterminés invariants dans une direction

We define an analogue of the Schur algorithm for transfer functions of lossless 2D systems which are invariant with respect to one of the variables. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On one-homogeneous solutions to elliptic systems in two dimensionsSur les solutions un-homogènes des systèmes elliptiques en dimension deux

In this Note we consider a class of nonlinear second order elliptic systems in divergence form and two independent variables. We prove that all Lipschitz continuous one-homogeneous weak solutions are linear. To cite this article: D. Phillips, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 39–42.     

The Bessel ratio distribution

Let X and Y be two random variables; then the exact distribution of the ratio $X/Y$ is derived when X and Y are independent Bessel function random variables. To cite this article: S. Nadarajah, S. Kotz, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Opérateurs de Fuchs non linéaires

In this Note, we study nonlinear partial differential equations of Fuchs type in spaces of functions sufficiently differentiable with respect to the Fuchsian variable and in Gevrey spaces with respect the other variables. The results are a generalization of those of Baouendi–Goulaouic obtained in the analytic case. To cite this article: P. Pongérard, C. Wagschal, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Tests de structure en régression sur variable fonctionnelle

In this Note we introduce a general approach to construct structural testing procedures in regression on functional variables. In the case of multivariate explanatory variables a well-known method consists in a comparison between a nonparametric estimator and a particular one. We adapt this approach to the case of functional explanatory variables. We give the asymptotic law of the proposed test statistic. The general approach used allows us to cover a large scope of possible applications as tests for no-effect, tests for linearity, …. To cite this article: L. Delsol, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Le problème de Cauchy local pour les plasmas dissipatifs

We investigate a system of partial differential equations modeling dissipative plasmas. Transport fluxes are anisotropic linear combinations of gradients and also include zeroth order contributions due to electromagnetic forces. There are also source terms depending on the solution gradient. By using entropic variables, we first recast the system in a partially symmetric form and next in the form of a quasilinear partially symmetric hyperbolic-parabolic system. Using a result of Vol'Pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem. To cite this article: V. Giovangigli, B. Graille, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Stability of sums of weighted nonnegative random variables

A stability result for sums of weighted nonnegative random variables is established and then it is utilized to obtain, among other things, a slight generalization of the Borel-Cantelli lemma and to show that the work of Jamison, Orey, and Pruitt (Z. Wahrsch. Verw. Gebiete4 (1965), 40–44) on almost sure convergence of weighted averages of independent random variables remains valid if the assumption of independence on the random variables is replaced by pairwise independence.     

Mesures quasi-invariantes pour un feuilletage et limites de moyennes longitudinales

In this Note, we generalize a result of Goodman–Plante, who characterizes limit points of averaging sequences as holonomy invariant transverse measures. We prove an analogous result for some leafwise averages, weighted with a cocycle Δ, whose limit points are a product of a quasi-invariant transverse measure with respect to Δ with a leafwise measure. To cite this article: B. Schapira, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sur un théorème de Kronecker concernant les variétés algébriques

A classical result of Kronecker, stated at the end of the Section 10 of Kronecker (J. Reine Angew. Math. 92 (1882) 1–123), is that any radical of a finitely generated ideal in a polynomial ring of n variables is the radical of an ideal generated by n+1 elements. We give a constructive and elementary proof of a generalisation presented in (Michigan Math. J. 31 (1984) (2) 167–180): in a ring of Krull dimension $\text{?}\text{n}$ a radical of a finitely generated ideal is the radical of an ideal generated by n+1 elements. To cite this article: T. Coquand, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Régression non-paramétrique pour des variables aléatoires fonctionnelles mélangeantesNonparametric regression for mixing functional random variables

This paper concerns a nonparametric regression model for dependent functional variables. The aim is to explain a r.r.v. Y by a functional regressor namely a variable X which is valued in some semi-normed vector space. A kernel type estimator is proposed and asymptotics with rates are proved under strong mixing assumption. To cite this article: F. Ferraty, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 217–220.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

Existence d'une sélection continue pour les applications multivoques à images non nécessairement convexes

In this Note, we prove a continuous selection existence result for the multivalud maps with not necessarily convex values, but continuously contractible, a notion that we define below. To cite this article: Y. Askoura, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Test asymptotiquement minimax pour une hypothèse nulle composite dans le modèle de densitéAsymptotically minimax test for a composite null hypothesis in the density model

Consider a sample of N random variables independent and identically distributed. Assume the density function belongs to a Hölder space. We construct an asymptotically minimax test and obtain the miximax rate of testing for the problem: the density function belongs to a parametric set versus the alternative: the distance in L₂[0,1] between the density function and the parametric set is bounded away from 0. To cite this article: C. Pouet, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 913–916.