# Intégrales stables multiples : propriétés des lois ; principe local d'invariance pour des variables aléatoires stationnaires

Abstract : In the first part, we study the laws of some stochastic integrals. After the introducing case of Poisson integrals for which we study the absolute continuity, we construct multiple stable integrals for functions in an Orlicz type space. To this way, we use a generalization of LePage representation. This representation is suitable to apply the stratification method and to study the laws of these integrals. We find in particular a condition ensuring absolute continuity of joint laws of multiple stable integrals with respect to the Lebesgue measure. We prove also from this representation the continuity for total variation norm of the laws of these integrals with respect to integrated functions. In the second part, we are interested in strong convergence of laws of stochastic functionals. We first consider a sequence $(\xi_n)_n$ of {\it i.i.d.} random variables and we associate processes of normalized partial sums. We get then interested in the convergence in variation of the laws of functionals of these processes to those of functionals of Wiener process. This type of convergence strengthens the ones of functional central limit theorem and allows to obtain local invariance principle. We prove such a convergence for a large class of functionals under hypothesis on the common law of the $\xi_n$'s weaker than those of the former results. We give real examples of such functionals for which these convergence holds. We show, to conclude, a similar result starting from some sequence of strongly dependent random variables. We obtain in this way, for example, a result of convergence in variation of the laws of normalized sums of dependent variables.
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https://tel.archives-ouvertes.fr/tel-00001343
Contributor : Jean-Christophe Breton <>
Submitted on : Wednesday, May 15, 2002 - 12:32:45 PM
Last modification on : Thursday, February 21, 2019 - 10:34:03 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:20:58 PM

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• HAL Id : tel-00001343, version 1

### Citation

Jean-Christophe Breton. Intégrales stables multiples : propriétés des lois ; principe local d'invariance pour des variables aléatoires stationnaires. Mathématiques [math]. Université des Sciences et Technologie de Lille - Lille I, 2001. Français. ⟨tel-00001343⟩

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