A Stochastic Contraction Mapping Theorem

07/01/2022

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by   Anthony Almudevar, et al.

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In this paper we define contractive and nonexpansive properties for adapted stochastic processes X_1, X_2, … 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.