Inégalités d'oracle pour l'estimation d'une densité de probabilité

We study the problem of the nonparametric estimation of a probability density in $L_2(\mathbb{R})$. Expressing the mean integrated squared error in the Fourier domain, we show that it is close to the mean squared error in the Gaussian sequence model. Then, applying a modified version of Stein's blockwise method, we obtain a linear monotone oracle inequality and a kernel oracle inequality. As a consequence, the proposed estimator is sharp minimax adaptive (i.e. up to a constant) on a scale of Sobolev classes of densities. To cite this article: Ph. Rigollet, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur la densité d'état de l'opérateur de Schrödinger quasi-périodique unidimensionnel

We prove two results on the density of states of the discrete one dimensional quasi-periodic Schrödinger equation with an analytic potential and Diophantine frequencies in the perturbed regime. On the one hand, we prove that this function has the behavior of a Hölder-$\frac{1}{2}$ function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. To cite this article: S. Hadj Amor, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Distribution asymptotique de l'affinité L2 de densités gaussiennes

After recalling the L² affinity measure between two multidimensionnal Gaussian density functions, we prove its asymptotic normality when parameters of one density are replaced by their maximum likelihood estimators. We extend this result to the vectorial case and we use it to allocate an estimated density to a class of densities as does the discriminant analysis method. As an application, a dating method for archeological data is proposed.     

Interprétation motivique de la formule d'excès d'intersection

Following Voevodsky, one can associate a Gysin triangle in the category of triangulated motives to every couple (X,Z) such that X is a smooth scheme and Z is a smooth closed subscheme of X. We study the functoriality of this triangle: if one considers a morphism $\text{f}\text{:Y→X}$ which is transversal to Z, this functoriality is simple. We are rather interested in the case where f is not quite transversal to Z. In that case, one obtains a motivic excess intersection formula and a ramification formula in the sense of equicharacteristic discrete valuation rings. To cite this article: F. Déglise, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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).     

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.     

Vitesse de convergence en norme p-intégrale et normalité asymptotique de l'estimateur crible de l'opérateur d'un ARB(1)

The autoregressive model in a Banach space (ARB) allows to represent many continuous time processes used in practice (see, for example, D. Bosq, Linear Processes in Function Spaces: Theory and Applications, 2000, Springer, p. 150). In this Note we study an estimator of the operator in ARB(1) by the least squares method, when the operator is strictly p-integral, $p\in ]1,\infty [$, and we use Grenander's method of sieves (From U. Grenander, Abstract Inference, Wiley, 1981). We show consistency of the sieve estimator and we derive a central limit theorem for this estimator. To cite this article: F. Rachedi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Lissité rationnelle des variétés de représentations d'un carquois

Recent works are devoted to the rational smoothness of varieties of representations for quivers, notably those of Robert Bédard, Ralf Schiffler and Philippe Caldéro. Their main result is the equivalence for a variety of representations $\text{O}$ of a quiver of Dynkin type, between

$\text{O}\text{is rationally smooth}\text{?}\text{O}\text{is smooth}\text{?}\text{O}\text{is an affine space.}$

This Note gives a different approach to the question and a strengthened version of Bédard–Schiffler–Caldéro's equivalences. To cite this article: A. Arabia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Estimation de la matrice de variance asymptotique des estimateurs du QMV de modèles ARMA faibles multivariés

In this Note, we consider the problems of estimating the asymptotic variance of the quasi-maximum likelihood estimator (QMLE) of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak VARMA). We first give expressions for the derivatives of the VARMA residuals in terms of the parameters of the models. Secondly we give an explicit expression of the asymptotic variance of the QMLE, in terms of the VAR and MA polynomials, and of the second- and fourth-order structure of the noise. We deduce a consistent estimator of the asymptotic variance of the QMLE.     

Comportement asymptotique à haute conductivité de l'épaisseur de peau en électromagnétisme

We study a three-dimensional model for the skin effect in electromagnetism. We first give a multiscale asymptotic expansion for the solution of the harmonic Maxwell equations set on a domain made of two materials, dielectric and highly conducting, with a regular interface between them. To measure the skin effect, we introduce a suitable skin depth function defined on the interface and generalizing the classical scalar quantity. We then prove an asymptotic expansion at high conductivity for this function, which exhibits the influence of the geometry of the interface on the skin depth.     

Continuité de la densité de l'intégrale d'aire dans les espaces de Hardy

Nous estimons la dépendance en la fonction harmonique des fonctionnelles densité de l'intégrale d'aire. Nous obtenons notamment un contrôle en norme de sup _aR D u(,a)–D v(,a) ainsi qu'une inégalité aux bons- de type exponentielle.     

Densité des zéros des transformés de Lévy itérés d'un mouvement brownien

The Lévy transform of a Brownian motion B is the Brownian motion $\text{B′}{}_{\mathrm{t}}\text{=∫}{}_0{}^{\mathrm{t}}\text{sgn}\text{B}{}_{\mathrm{s}}\text{d}\text{B}{}_{\mathrm{s}}$; denote by Bⁿ the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that B_tⁿ=0 for some n, is a.s. dense in the time-axis $\text{R}{}_{\mathrm{+}}$. To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Singularité de la mesure harmonique sur le bord d'un groupe hyperbolique

This Note deals with the dimension of the harmonic measure ν associated with a random walk on the isometry group of a Gromov hyperbolic space. We establish a link of the form $\dim \nu ?h/l$ between the dimension of the harmonic measure, the asymptotic entropy h of the random walk and its rate of escape l. Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure. To cite this article: V. Le Prince, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Processus de Bickel–Rosenblatt pondéré et tests d'ajustement

The goal of this work is to establish the limit distribution of the process

$\text{I}{}_{\mathrm{n}}\text{(W):=}\text{∫}\text{A}\text{f}{}_{\mathrm{n}}\text{(x)?Ef}{}_{\mathrm{n}}\text{(x)}{}^2\text{W(x)}\text{d}\text{x,}\text{W∈}\text{W}\text{,}$

where $\text{W}$ is a class of weight functions W, f_n is the kernel density estimator of the density f and A is a Borelian subset of $\text{R}$. We apply this result to derive new statistics to test goodness-of-fit of the density function f. Under some local alternatives, these new tests are more powerful than the usual Bickel–Rosenblatt one. To cite this article: F. Chebana, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Un problème d'optimisation de forme pour la contrôlabilité exacte de l'équation des ondes 2D

We consider the wave equation defined on $\Omega ?{\mathbb{R}}^2$ and $\omega ?\Omega$. We designate by $v_{\omega }$ the distributed control of minimal $L^2(\omega \times (0,T))$ norm obtained with the Hilbert Uniqueness Method which stabilizes the system at time $T>0$. This Note addresses the question of the optimal position of ω in order to minimize $J:\omega \to 6v_{\omega }{6}_{L^2(\omega \times (0,T))}$. Assuming $\omega \in C^{1,1}(\Omega )$, we express the shape derivative of J as a curvilinear integral on ?ω (independently of any adjoint solution) leading to a descent algorithm. A numerical application is given. To cite this article: A. Münch, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Sur la possibilité d'appliquer la méthode de Riemann aux séries trigonométriques sommables par le procédé de Poisson

Sans résuméRédigé par A. Zygmund.     

L'inégalité de Bohr pour les séries de Dirichlet

We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if $f(s)={\sum }_{n=1}^{\infty }{a}_n{n}^{?s}$, with $6f{6}_{\infty }:={\sup }_{\mathfrak{R}s>0}|f(s)|<\infty$, then ${\sum }_{n=1}^{\infty }|a_n|n^{\mathrm{?2}}?6f{6}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }|a_n|n^{?1/2}?C6f{6}_{\infty }$, C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Solutions globales d'énergie infinie de l'équation de Navier–Stokes 2D

We study in this Note the solutions of the 2D Navier–Stokes equations with initial data in ?BMO. For $u{|}_{t=0}$ in the closure of the Schwartz class, we obtain the existence and uniqueness of a global solution, and besides an estimate on its norm in ?BMO. To cite this article: P. Germain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).