An iterated random function with Lipschitz number one

Abstract

Consider the set of functions $f_{\hat{I}}¸(x)=|\hat{I}¸-x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0\in \mathbb{R}$ and continues with $x_{k+1}=f_{\hat{I}{¸}_{k+1}}(x_k)$ with each $\hat{I}{¸}_{k+1}$ picked from a fixed bounded distribution $\hat{I}¼$ on ${\mathbb{R}}^{+}$. We prove the conjecture of G. Letac that if $\hat{I}¼$ is not supported on a lattice, then this process has a unique stationary distribution $\ddot{I}{€}_{\hat{I}}¼$ and any distribution converges under iteration to $\ddot{I}{€}_{\hat{I}}¼$ (in the weak-${}^{\ast }$ topology). We also give a bound on the rate of convergence in the special case that $\hat{I}¼$ is supported on a two-point set. We hope that the techniques will be useful for the study of other Markov processes where the transition functions have Lipschitz number one.

This is the eighth of eleven old articles being uploaded to arxiv after publication