Random iteration with place dependent probabilities

Abstract

Markov chains arising from random iteration of functions $S_{\theta}:X\to X$, $\theta \in \Theta$, where $X$ is a Polish space and $\Theta$ is arbitrary set of indices are considerd. At $x\in X$, $\theta$ is sampled from distribution $\theta_x$ on $\Theta$ and $\theta_x$ are different for different $x$. Exponential convergence to a unique invariant measure is proved. This result is applied to case of random affine transformations on ${\mathbb R}^d$ giving existence of exponentially attractive perpetuities with place dependent probabilities.