Consider the set of functions $f_{ \theta}(x)=|\theta -x|$ on $\mathbb{ R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{ R}$ and continues with $x_{ k+1}=f_{ \theta_{ k+1}}(x_{ k})$ 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-$^*$ 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.
Categories: math.PR math.DS