Indépendance conditionnelle et uniformité pour les lois fortes des grands nombres dans les espaces de Banach

Summary A. Beck has given an uniform strong law of large numbers for families of mutually symmetric and uniformly essentially bounded sequences of centered random variables, with values in (k, )—B-convex spaces. We show that, without any limitation on the Banach spaces, the technique used by A. Beck allows to replace, in strong law of large numbers making use of conditions bearing on essential bounds, the hypothesis of independence by an hypothesis called conditional-independence-and-centering, which is weaker than both hypothesis of independence and of mutual symmetry; moreover, in several cases, one gets uniform strong laws of large numbers (for families of conditionally-independent-and-centered sequences). The results we get are compared with recent results of G. Pisier, obtained with type p spaces techniques.     

Convergence presque sûre de l'algorithme de Kohonen unidimensionnel

We prove the a.s. convergence of the one dimensional Kohonen algorithm after selforganization and with decreasing step. Under some mild asumption of log-concavity on the input probability distribution, we first show the uniqueness of the limit and then the a.s. convergence. The proof relies on the index formula for a vector field on a manifold and a result by M. Hirsh about the convergence of cooperative dynamical systems applied to the ODE.     

Une version presque sûre du théorème de la limite centrale pour les algorithmes stochastiques

We prove an almost sure central limit theorem for some multidimensional stochastic algorithms used for the search of zeros of a function, and known to satisfy a central limit theorem.     

Résultats de convergence presque sûre pour l’estimation et la prévision des processus linéaires hilbertiens

Motivated by the prediction of certain types of continuous time processes, we study, here the estimation and the prediction of hilbertian linear processes. Only results about almost sure convergence are given.     

Dimension de Hausdorff du graphe d'une fonction continue : nouvelles majorations déterministes et minorations presque sûres

In this Note, we establish an upper bound of the Hausdorff dimension of the graph of a continuous function depending on its wavelet coefficients in a sufficiently regular wavelet basis. In particular, this bound is related to the Besov smoothness of the function. An almost sure lower bound of the same quantity is stated for some precise classes of random functions.     

Un théorème central limite presque sûr à moments généralisés pour les rotations irrationnelles

In a recent work, we indicated another formulation of the Almost Sure Central Limit Theorem (A.S.C.L.T.), with series in place of averages, by showing that the property of the A.S.C.L.T. directly follows from the theory of orthogonal sums. For, we used the notion of quasi-orthogonal systems introduced earlier by R. Bellmann, and later developed by Kac–Salem–Zygmund. The main object of this paper is to prove a similar result for irrational rotations of the torus. We prove the existence of a generalized moment version of the A.S.C.L.T., with a speed of convergence. In our strategy, we use again the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context. The proof avoid notably the use of Volny's result on the existence of good Gaussian approximations in aperiodic dynamical systems, and should also permit to be able to treat problems of comparable nature, in particular in non-ergodic cases. Received: 2 February 1999     

Le théorème limite central pour des sommes de Riesz-Raikov

Summary Let be an algebraic number greater than 1 andf a real 1-periodic function; ifF _N denotes the random variable defined on [0, 1] byF _N (t) , it is proved here that under sufficiently broad assumptions onf: 1) the sequence converges to a finite ²(0); 2) if >0, the sequence {F _N } converges in law to . We give an explicit computation of with respect to and a characterisation of functions for which =0.(Our results are also valid for almost every real >1).     

Vitesse de convergence presque sûre de l'estimateur à noyau du maximum de vraisemblance local

We consider the local maximum likelihood estimation of $\theta (x)$, unknown parameter of the conditional distribution of Y given $X=x$. The aim of this Note is the study of strong uniform consistency rates of the local maximum likelihood kernel estimator. Under suitable regularity conditions, we establish a uniform law of the logarithm for the maximal deviation of this estimator. The method of proof is based upon functional limit laws derived by modern empirical process theory. To cite this article: D. Blondin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

La singularité mutuelle presque sûre du spectre des transformations d’Ornstein

It is shown that, almost surely with respect to the product measure, the maximal spectral type of Ornstein’s transformations are mutually singular so that these transformations are mutually disjoint in Furstenberg’s sense.