A Stochastic Contraction Mapping Theorem

Abstract

In this paper we define contractive and nonexpansive properties for adapted stochastic processes $X_1, X_2, \ldots $ which can be used to deduce limiting properties. In general, nonexpansive processes possess finite limits while contractive processes converge to zero $a.e.$ Extensions to multivariate processes are given. These properties may be used to model a number of important processes, including stochastic approximation and least-squares estimation of controlled linear models, with convergence properties derivable from a single theory. The approach has the advantage of not in general requiring analytical regularity properties such as continuity and differentiability.