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We study the convergence of random function iterations for finding aninvariant measure of the corresponding Markov operator. We call the problem offinding such an invariant measure the stochastic fixed point problem. Thisgeneralizes earlier work studying the stochastic feasibility problem, namely,to find points that are, with probability 1, fixed points of the randomfunctions. When no such points exist, the stochastic feasibility problem iscalled inconsistent, but still under certain assumptions, the more generalstochastic fixed point problem has a solution and the random functioniterations converge to an invariant measure for the corresponding Markovoperator. We show how common structures in deterministic fixed point theory canbe exploited to establish existence of invariant measures and convergence indistribution of the Markov chain. This framework specializes to manyapplications of current interest including, for instance, stochastic algorithmsfor large-scale distributed computation, and deterministic iterative procedureswith computational error. The theory developed in this study provides a solidbasis for describing the convergence of simple computational methods withoutthe assumption of infinite precision arithmetic or vanishing computationalerrors.

Nonexpansive Markov Operators and Random Function Iterations for Stochastic Fixed Point Problems