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May 2015 On unbounded invariant measures of stochastic dynamical systems
Sara Brofferio, Dariusz Buraczewski
Ann. Probab. 43(3): 1456-1492 (May 2015). DOI: 10.1214/13-AOP903


We consider stochastic dynamical systems on $\mathbb{R}$, 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 $\mathbb{R}$. 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 $\mathbb{E}\log A_{1}=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. 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.


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Sara Brofferio. Dariusz Buraczewski. "On unbounded invariant measures of stochastic dynamical systems." Ann. Probab. 43 (3) 1456 - 1492, May 2015.


Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1352.37155
MathSciNet: MR3342668
Digital Object Identifier: 10.1214/13-AOP903

Primary: 37Hxx , 60J05
Secondary: 60B15 , 60K05

Keywords: affine recursion , invariant measure , Poisson equation , reflected random walk , stochastic dynamical system , stochastic recurrence equation

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • May 2015
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