A Stochastic Contraction Mapping Theorem

The submartingale convergence theorem is often described as a stochastic analogue to the bounded convergence theorem for monotone sequences of real numbers. We define an alternative process which is a stochastic analogue to an iterative contraction process. A general convergence theorem is presented, with an extension to multivariate processes. The convergence theorem functions in a manner similar to Kronecker’s lemma. Given contraction properties imposed on the conditional expectations the convergence of a random sequence can be verified by establishing the convergence of a related random series. In the examples presented in this article these random series possess the martingale property. Examples of processes which satisfy this definition include the sample mean process of a sequence of martingale differences, multivariate linear least squares estimators and the Robbins-Monro stochastic approximation algorithm. Supported by a grant from the Natural Sciences and Engineering Research Council AMS 2000 subject classifications. Primary 60G48; secondary 65C40.