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

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**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