  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.
