An iterated random function with Lipschitz number one

TL;DR

This paper proves the conjecture that a specific iterated random function process with Lipschitz number one has a unique stationary distribution when the distribution is not lattice-supported, and provides convergence rate bounds.

Contribution

It establishes the existence and uniqueness of the stationary distribution for a class of Lipschitz-one Markov processes, confirming a conjecture by G. Letac.

Findings

Unique stationary distribution exists under non-lattice support

Convergence to the stationary distribution is proven

Bound on convergence rate for two-point support case

Abstract

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.