Ergodic theorems for iterated function systems controlled by regenerative sequences
type: Item
instance of: scholarly article
title: Ergodic theorems for iterated function systems controlled by regenerative sequences
zbMATH Open document ID: 0920.28015
DOI: 10.1023/A:1022642328845
author: Örjan Stenflo
author: D. S. Silvestrov
published in: Journal of Theoretical Probability
publication date: 1999-07-20T00:00:00Z
review text: In the paper under review are considered special iterated function systems (IFS) on Polish spaces. Namely, if \((X,d)\) is a Polish space, \(S\) a measurable space and \(w:X\times S\to X\) a measurable function such that for any \(s\in S\), \(w_s\) is a continuous mapping, then the family \((X,w_s)_{s\in S}\) is called an IFS on \(X\). A stochastic sequence \((I_n)_{n\in\mathbb{N}}\) in \(S\) defines a stochastic dynamical system \(Z_n\) such that for any \(x_0\in X\), \(Z_0(x_0)=x_0\), and \[ Z_n(x_0)=w_{I_{n-1}}\circ w_{I_{n-2}}\circ\cdots\circ w_{I_0}. \] This type of stochastic dynamical system is called an IFS controlled by the stochastic sequence \((I_n)\). In the present paper are investigated IFS controlled by a regenerative process with discrete time and arbitrary state space. Mainly, the authors establish ergodic theorems of distributional and large numbers types which are uniform with respect to initial points chosen in compact sets. The asymptotic behaviour for the distribution of the random vector \((Z_n(x_0),I_n)\) is also determined.
Mathematics Subject Classification ID: 28D99
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 1208773
zbMATH Keywords: ergodic theorem
zbMATH Keywords: iterated function system
zbMATH Keywords: stochastic dynamical system
zbMATH Keywords: regenerative process
zbMATH Keywords: IFS
reviewed by: Emilia Petrisor
MaRDI profile type: Publication
full work available at URL: a:1022642328845
OpenAlex ID: W1590059880
description: scientific article; zbMATH DE number 1208773