ON UNBOUNDED INVARIANT MEASURES OF STOCHASTIC DYNAMICAL SYSTEMS

• 成果类型: Article
• 署名作者: Brofferio, Sara; Buraczewski, Dariusz
• 署名单位: Universite Paris Saclay; Universite Paris Saclay; University of Wroclaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP903
• 发表日期: 2015
• 页码: 1456-1492
• 关键词: equation x-n; weak contractivity; limit-theorem; random-walks; recursions

摘要

We consider stochastic dynamical systems on R, that is, random processes defined by X-n(x) = Psi(n)(X-n-1(x)), X-0(x) = x, where Psi(n) are i.i.d. random continuous transformations of some unbounded closed subset of R. We assume here that Psi(n) behaves asymptotically like A(n)x, for some random positive number A(n) [the main example is the affine stochastic recursion Psi(n) (x) = A(n)x + B-n]. Our aim is to describe invariant Radon measures of the process X-n(x) in the critical case, when E log A(1) = 0. We prove that those measures behave at infinity like dx/x. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval [0, 1], additive Markov processes and a variant of the Galton-Watson process.