A Fixed Point Theorem for Iterative Random Contraction Operators over Banach Spaces

Consider a contraction operator T over a Banach space X with a fixed point x. Assume that one can approximate the operator T by a random operator T̂ using N ∈ N independent and identically distributed samples of a random variable. Consider the sequence (X̂ k )k∈N, which is generated by X̂ N k+1 = T̂ (X̂ k ) and is a random sequence. In this paper, we prove that under certain conditions on the random operator, (i) the distribution of X̂ k converges to a unit mass over x as k and N goes to infinity, and (ii) the probability that X̂ k is far from x ⋆ as k goes to infinity can be made arbitrarily small by an appropriate choice of N . We also find a lower bound on the probability that X̂ k is far from x ⋆ as k → ∞. We apply the result to study probabilistic convergence of certain randomized optimization and value iteration algorithms.