A Stochastic Contraction Mapping Theorem

TL;DR

This paper introduces a stochastic contraction mapping theorem that characterizes the convergence of adapted stochastic processes using contractive and nonexpansive properties, applicable to various stochastic models without requiring regularity conditions.

Contribution

It develops a unified theoretical framework for analyzing convergence of stochastic processes based on contraction properties, extending to multivariate cases and broad applications.

Findings

Nonexpansive processes have finite limits.

Contractive processes converge to zero almost surely.

Applicable to stochastic approximation and linear model estimation.

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.