Iterated function systems with place dependent probabilities and application to the Diaconis-Friedman's chain on [0,1]

TL;DR

This paper investigates Markov chains generated by iterated Lipschitz function systems with place-dependent probabilities, establishing conditions for unique invariant measures and analyzing the Diaconis-Friedman's chain on [0,1].

Contribution

It introduces a general framework for analyzing such Markov chains and applies the method to a specific chain on [0,1], extending previous results.

Findings

Proves uniqueness of invariant measures under broad conditions

Uses quasi-compact operator techniques for analysis

Describes the behavior of the Diaconis-Friedman's chain with place-dependent probabilities

Abstract

We study Markov chains generated by iterated Lipschitz functions systems with possibly place dependent probabilities. Under general conditions, we prove uniqueness of the invariant probability measure for the associated Markov chain, by using quasi-compact linear operators technics. We use the same approach to describe the behavior of the Diaconis-Friedman's chain on [0,1] with possibly place dependent probabilities.