The auto-regressive model
on $\RR ^{d}$ defined by the recurrence equation
$ Y^{y}_{n}=a_{n}Y^{y}_{n-1}+B_{n} $,
where $ \{ (a_{n},B_{n})\} _{n} $\vspace*{-0.5pt} is a sequence of i.i.d.
random variables in $ \RR ^{*}_{+}\times \RR ^{d} $, has, in the
critical case $ \esp {\log a_{1}}=0 $,\vspace*{-0.5pt} a local
contraction property,
that is, when $ Y^{y}_{n} $ is in a compact set the distance
$ | Y^{y}_{n}-Y^{x}_{n}| $ converges almost surely to 0.
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 autoregressive model whose linear part
is given by triangular matrices with first Lyapounov exponent equal
to 0. 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.