  We study the convergence of random function iterations for finding an
invariant measure of the corresponding Markov operator. We call the problem of
finding such an invariant measure the stochastic fixed point problem. This
generalizes earlier work studying the stochastic feasibility problem}, namely,
to find points that are, with probability 1, fixed points of the random
functions [Hermer, Luke, Sturm, 2019]. When no such points exist, the
stochastic feasibility problem is called inconsistent, but still under certain
assumptions, the more general stochastic fixed point problem has a solution and
the random function iterations converge to an invariant measure for the
corresponding Markov operator. There are two major types of convergence: almost
sure convergence of the iterates to a fixed point in the case of stochastic
feasibility, and convergence in distribution more generally. We show how common
structures in deterministic fixed point theory can be exploited to establish
existence of invariant measures and convergence of the Markov chain. We show
that weaker assumptions than are usually encountered in the analysis of Markov
chains guarantee linear/geometric convergence. This framework specializes to
many applications of current interest including, for instance, stochastic
algorithms for large-scale distributed computation, and deterministic iterative
procedures with computational error. The theory developed in this study
provides a solid basis for describing the convergence of simple computational
methods without the assumption of infinite precision arithmetic or vanishing
computational errors.
