Systems of random mappings with a common fixed point

Mappings from a set of Lipschitz-continuous mappings are applied successively on an initial point x ₀ in a complete metric space. All mappings possess the same fixed point x and are applied at random with repetitions. A lower bound is found for the probability that the system’s state after n iterations, x_n, is within a p-neighbour-hood of the common fixed point x. Moreover, sufficient conditions on the Lipschitz constants and on the probabilities of occurrence of the mappings which guaranteee convergence of the system’s state to x in the mean-square sense are found.