Ergodic theorems for Markov chains represented by iterated function systems (Q2725205)

Using standard coupling of an iterated function system with its backward iterates, this paper proves several theorems about Markov chains represented by iterated function systems. The first is a slight generalization of a result by \textit{J. H. Elton} [Stochastic Processes Appl. 34, No. 1, 39-47 (1990; Zbl 0686.60028)], showing that when the iterated function system satisfies the average contractivity condition \(Ed(w(x),w(y))\leq cd(x,y)\) for some \(c<1\) (where \(w\) is the random function being iterated), then the Markov chain converges to a unique invariant probability distribution. A bound on the distance between the invariant distribution and the distribution at step \(n\) is derived as well. NEWLINENEWLINENEWLINEMore novel is another application to parametrized families of iterated function systems, computing bounds on the distance between the invariant distributions of two chains in terms of the behavior of the representing functions. The paper also combines the ergodicity of iterated function systems with known results on the representation of Markov chains as iterated function systems, to provide a new proof of the geometric ergodicity of Markov chains satisfying the Doeblin splitting condition. Several applications are given, as well, to iterated function systems with place-dependent probabilities.