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 HAL : hal-00456818, version 1
 arXiv : 0809.4349
 Probability Theory and Related Fields 148, 3-4 (2010) 333-402
 Convergence to stable laws for a class of multidimensional stochastic recursions
 (2010)
 We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that $M_n$ takes values in the similarity group of $\R^d$, and the Markov chain has a unique stationary measure $\nu$, which has unbounded support. We denote by $|M_n|$ the expansion coefficient of $M_n$ and we assume $\E |M|^\a=1$ for some positive $\a$. We show that the partial sums $S_n=\sum_{k=0}^n X_k$, properly normalized, converge to a normal law ($\a\ge 2$) or to an infinitely divisible law, which is stable in a natural sense ($\a<2$). These laws are fully nondegenerate, if $\nu$ is not supported on an affine hyperplane. Under a natural hypothesis, we prove also a local limit theorem for the sums $S_n$. If $\a\le 2$, proofs are based on the homogeneity at infinity of $\nu$ and on a detailed spectral analysis of a family of Fourier operators $P_v$ considered as perturbations of the transition operator $P$ of the chain $\{X_n \}$. The characteristic function of the limit law has a simple expression in terms of moments of $\nu$ ($\a > 2$) or of the tails of $\nu$ and of stationary measure for an associated Markov operator ($\a\le 2$). We extend the results to the situation where $M_n$ is a random generalized similarity.
 1 : Instytut Matematyczny Uniwersytet Wroclawski 2 : Institut de Recherche Mathématique de Rennes (IRMAR) CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
 Domaine : Mathématiques/Probabilités
 Mots Clés : probability
 Lien vers le texte intégral : http://fr.arXiv.org/abs/0809.4349
 hal-00456818, version 1 http://hal.archives-ouvertes.fr/hal-00456818 oai:hal.archives-ouvertes.fr:hal-00456818 Contributeur : Marie-Annick Guillemer <> Soumis le : Lundi 15 Février 2010, 17:58:43 Dernière modification le : Lundi 15 Novembre 2010, 16:39:22