An iterated random function with Lipschitz number one

Consider the set of functions f_θ(x) = |θ - x| on R. Define a, Markov process that starts with a point x₀ ∈ R. and continues with X_k+1 = f_θk+1 (x_k) with each θ_k+1 chosen from a fixed bounded distribution μ on R⁺. We prove the conjecture of Letac that if μ is not supported on a lattice, then this process has a unique stationary distribution π_μ and any distribution converges under iteration to π_μ (in the weak-* topology). We also give a bound on the rate of convergence in the special case that μ 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.