Systèmes quasi-orthogonaux et théorème central limite presque sur

We prove, in the case of irrational rotations, the existence of a generalized moment version of the Almost Sure Central Limit Theorem (ASCLT), with a speed of convergence. In our strategy, we use the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context, which should also permit to be able to treat problems of comparable nature.     

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.     

Distributions diophantiennes et théorème limite local sur

We study the speed of convergence of n^d/2fd^*n in the local limit theorem on under very general conditions upon the function f and the distribution . We show that this speed is at least of order and we give a simple characterization (in diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough f. We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when f is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks.     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Un théorème sur les anneaux primaires et complètement intégralement clos

Sans résumé     

Un Théorème sur les Espaces homogènes complexes

Sans résuméDédié à M. lc Prof.Alexandre Ostrowski     

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).     

Sur le théorème limite du calcul des probabilités

Sans résumé     

Un théorème de dualité

Sans résumé Herrn H. Hopf zum sechzigsten Geburtstag gewidmet     

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     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Un théorème de représentation pour les martingales discontinues

We consider a fixed integer-valued random measure (also called a random point process). We represent any local martingale as the sum of a stochastic integral with respect to this random measure, and of a local martingale which does not jump on the support of the random measure.     

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.