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October 2003 Speed of stochastic locally contractive systems
Sara Brofferio
Ann. Probab. 31(4): 2040-2067 (October 2003). DOI: 10.1214/aop/1068646377


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.


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Sara Brofferio. "Speed of stochastic locally contractive systems." Ann. Probab. 31 (4) 2040 - 2067, October 2003.


Published: October 2003
First available in Project Euclid: 12 November 2003

zbMATH: 1046.60067
MathSciNet: MR2016611
Digital Object Identifier: 10.1214/aop/1068646377

Primary: 60J10
Secondary: 60B12 , 60B15 , 60G50

Keywords: Contractive system , iterated functions system , limit theorems , Random coefficients autoregressive model , Random walk , stability

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 4 • October 2003
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