An iterated random function with Lipschitz number one

Consider the set of functions $f_{\theta }(x)=\mathrm{\mid }\theta -x\mathrm{\mid }$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0\in \mathbb{R}$ and continues with $x_{k+1}=f_{{\theta }_{k+1}}(x_k)$ with each ${\theta }_{k+1}$ picked from a fixed bounded distribution $\mu$ on ${\mathbb{R}}^{+}$. We prove the conjecture of G. Letac that if $\mu$ is not supported on a lattice, then this process has a unique stationary distribution ${\pi }_{\mu }$ and any distribution converges under iteration to ${\pi }_{\mu }$ (in the weak-${}^{\ast }$ topology). We also give a bound on the rate of convergence in the special case that $\mu$ 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.