Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Stochastic locally contractive systems with speed

Auteur(s):

Code(s) de Classification MSC:

• 60J27 Markov chains with continuous parameter
• 60B15 Probability measures on groups, Fourier transforms, factorization
• 60J15 Random walks

Résumé: The auto-regressive model on $$\RR ^{d}$$ defined by the recurrence equation $$Y^{y}_{n}=a_{n}Y^{y}_{n-1}+B_{n}$$,where $$\left\{ (a_{n},B_{n})\right\} _{n}$$ is a sequence of i.i.d. random variables in $$\RR ^{*}_{+}\times \RR ^{d}$$ has, in the critical case $$\esp {\log a_{1}}=0$$, a local contraction property, i.e. when $$Y^{y}_{n}$$ is in a compact set the distance $$\left| Y^{y}_{n}-Y^{x}_{n}\right|$$ converges almost surely to zero. We determine the speed of this convergence and we use this asymptotic estimate to deal with some higher dimensional situations. In particular we prove the recurrence and the local contraction property with speed for an auto-regressive model whose linear part is given by triangular matrices with first Lyapounov exponent equal to zero. We extend the previous results to a Markov chain on a nilpotent Lie group induced by a random walk on a solvable Lie group of $$\mathcal{NA}$$ type.

Mots Clés: Random coefficients auto-regressive model ; Limit theorems ; Stability ; Random walk ; Contractive system ; Iterated function system

Date: 2002-03-29

Prépublication numéro: PMA-719

Postscript file : PMA-719.ps

Pdf file : PMA-719.pdf