Systems of random iterative continuous mappings with a common fixed point

A study is made of systems of Lipschitz-continuous mappings which are applied successively on an initial point x₀ in a closed nonempty complete metric space. All mappings possess the same fixed point x*, and are applied at random with repetitions by choosing a mapping from a finite set of such functions. A lower bound is found on the probability that the system´s state after n iterations, x_n, is within a ρ-neighborhood of the common fixed point, x*. Conditions are developed that guarantee that the lower bound is (eventually) monotonically increasing in n