Speed of stochastic locally contractive systems

• 成果类型: Article
• 署名作者: Brofferio, S
• 署名单位: Graz University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1068646377
• 发表日期: 2003
• 页码: 2040-2067
• 关键词: affine group; random-walk

摘要

The auto-regressive model on R-d defined by the recurrence equation Y-n(y) = a(n)Y(n-1)(y) + B-n, where {(a(n), B-n)}(n) is a sequence of i.i.d. random variables in R+* x R-d, has, in the r=critical case E[loga(1)] = 0, a local contraction property, that is, when Y-n(y) is in a compact set the distance \Y-n(y) - Y-n(x)\ 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 NA type.