On a theorem of Dubins and Freedman (Q1970317)

The authors extend the Dubins-Freedman theorem on existence and uniqueness of an invariant measure of a Markov process generated by i.i.d. continuous monotone random maps of an interval and on exponential fast convergence to it of transition probabilities. They prove first a general result for i.i.d. maps of an abstract measurable space under a general contraction and ``splitting'' condition assuming that probability measures there form a complete metric space with respect to the variational distance. Then they derive a generalization of the Dubins-Freedman theorem as a corollary discarding the continuity condition. They obtain also several other results in this direction and consider an example of iterations of two quadratic maps chosen at random.