Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Stochastic locally contractive systems with speed


Code(s) de Classification MSC:

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