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In this paper, we consider the problem of finding an almost surely commonfixed point of a family of paracontraction maps indexed on a probability space,which we refer to as the stochastic feasibility problem. We show that a randomiteration of paracontraction maps driven by an ergodic stationary sequenceconverges, with probability one, to a solution of the stochastic feasibilityproblem, provided a solution exists. As applications, we obtain non-white noiserandomized algorithms to solve the stochastic convex feasibility problem andthe problem of finding an almost surely common zero of a collection of maximalmonotone operators.

Random iterations of paracontraction maps and applications to feasibility problems