Estimation des ordres de modèles ARMA faibles multivariés

In this Note, we consider the problem of order selection of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated, but not necessarily independent. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be iid. This selection is based on minimizing an information criterion, especially that introduced by Akaike. The theoretical foundations of the Akaike information criterion (AIC) are not more established when the iid assumption on the noise is relaxed. We propose a modified AIC criterion, and which may be very different from the standard AIC criterion.     

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.     

Estimation de la précision asymptotique dans l'estimation de modèles ARMA faibles

An estimator of the asymptotic covariance matrix of the least squares estimator of ARMA models is proposed. To deal with the ARMA representations of nonlinear processes, the standard strong assumptions on the noise are replaced by a mixing condition on the observed process. The weak consistency of the proposed estimator is proved.     

Modèles semi-factoriels et modèles de Néron

Let S be the spectrum of a discrete valuation ring with function field K. Let X be a scheme over S. We will say that X is semi-factorial over S if any invertible sheaf on the generic fiber X _K can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a proper, flat, normal and semi-factorial model over S. We also construct some semi-factorial compactifications of regular S-schemes, such as Néron models of abelian varieties. The semi-factoriality property for a scheme X/S corresponds to the Néron property of its Picard functor. In particular, one can recover the Néron model of the Picard variety ${{\rm Pic}_{X_K/K,{\rm red}}^0}$ of X _K from the Picard functor Pic _X/S , as in the known case of curves. This provides some information on the relative algebraic equivalence on the S-scheme X.     

Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus

The work we present here is focused on the justification of continuum mechanics as a natural asymptotic of molecular theories when the interatomic distance goes to zero. We give a sound theoretical ground to existing models, and derive and next study (apparently) new ones.     

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.     

Sur l'estimation des modèles autorégressifs d'ordre multiple de séries temporelles

In this Note, we determine the minimum Hellinger distance estimate of a multiple order autoregressive time series model. Under some assumptions which ensure some probabilistic properties, and under other mild conditions, we establish the asymptotic properties of this estimate. As an example, we consider the exponential autoregressive models.     

Sur les transformations de Schlesinger de la sixième équation de Painlevé

Following the recent discovery of two new Schlesinger transformations (ST) for the sixth Painlevé equation, we give the interrelations between all the known STs. We thus isolate the unique one which at the same time conserves the independent variable and is not a product of other STs.     

Convergence d'un modèle à vitesses discrètes pour l'équation de Boltzmann-BGK

We consider a conservative and entropie discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. First, we prove a consistency result for this approximation. Then, we demonstrate that the discrete-velocity model possesses a unique solution. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.     

Principes de déviations modérées pour des modèles autorégressifs d'ordre p

We establish Moderate deviations principles (MDP) for some stable autoregressive models of order p: first for continuous unbounded additive functionnals of this process, second for the estimation error of the regression function (least-squares estimator in the stable linear case, kernel estimator in the Lipschitz non-linear case).     

Sur les intégrales rationnelles de l'équation de Kummer

Sans résumé     

Sur l'existence et l'unicité de solutions de modèles linéaires en aéroacoustique

An aeroacoustic modelling is considered in this Note. A steady flow modifies the classical wave propagations. But using physical assumptions on the data, it is proved an existence and uniqueness result. Furthermore, practical examples prove that one cannot get rid of the required hypothesis.     

Inférence fondée sur la vraisemblance pour des modèles de records

In a time series $\{X_t,t\ge 1\}$, $X_j$ is said to be an upper record if $X_j>\max ?\{X_1,\dots ,X_{j?1}\}$. Some popular models for records are the Yang–Nevzorov and the Linear Drift models. In this note, we introduce for these models the joint likelihood of the record sequence and the indicators of their occurrence. This likelihood can then be used to obtain estimators of the unknown parameters in the models. It can also be used to derive inferential procedures associated with the selection of a proper model for such data.     

Critères d'ergodicité géométrique ou arithmétique de modèles linéaires pertubés à représentation markovienne

This paper is devoted to some models with a state space representation ( 1 ). Foster-Lyapounov type inequalities due to Tweedie [11], Tuominen and Tweedie [10] describe the conditions under which such a model remains ergodic. The applications are twofold. In the first case we examine an autoregressive model which departs from linearity additively. In the second case, the state space model arises from a sublinear model. In each case conditions that ensure ergodicity are given, with geometric or, possibly, subgeometric rates of convergence to the ergodic measure.     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

Quelques modèles asymptotiques pour les poutres purement encastrées

We present several limit models for genuinely clamped elastic beams obtained from linear three-dimensional elasticity using an asymptotic method, and we prove the existence and the uniqueness of the solution for these models. We also obtain convergence to the usual clamped beam model.     

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.     

Tests d'hypothèses sur les coefficients de Fourier d'une fonction de régression non linéaire

We consider the nonparametric regression model with regression function ƒ. We construct tests for hypotheses on Fourier coefficients of ƒ in a given band of frequencies. Such tests can be used in particular to compare two noisy signals in a frequency band. The test statistics, we use, are based on the empirical Fourier coefficients of the regression function ƒ.