Speed of stochastic locally contractive systems

Abstract

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.