On the iterations of some random functions with Lipschitz number one

For the iterations of $x\mapsto |x-\theta|$ random functions with Lipschitz number one, we represent the dynamics as a Markov chain and prove its convergence under mild conditions. We also demonstrate that the Wasserstein metric of any two measures will not increase after the corresponding induced iterations for measures and identify conditions under which a polynomial convergence rate can be achieved in this metric. We also consider an associated nonlinear operator on the space of probability measures and identify its fixed points through an detailed analysis of their characteristic functions.