A strong law of large numbers for nonexpansive vector-valued stochastic processes

Abstract

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, $${\mathcal{F}}_0 \subset {\mathcal{F}}_1 \subset ... \subset {\mathcal{F}}_n $$ an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly $${\mathcal{F}}_n $$ -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if $$E(x_{n + 1} \left| {{\mathcal{F}}_n ) = T(x_n )} \right.$$ . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (xn) be aT-martingale taking on values inH. If $$\sum\limits_{n = 1}^\infty {n^{ - 2} } E\left\| {x_{n + 1} - Tx_n } \right\|^2< \infty ,$$ then x n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (xn) be aT-martingale (taking on values inX). If $$\sum n ^{ - p} E\left( {\left\| {x_n - Tx_{n - 1} } \right\|^p } \right)< \infty ,$$ then there exists a continuous linear functionalf∈X * of norm 1 such that $$\mathop {\lim }\limits_{n \to \infty } f(x_n )/n = \mathop {\lim }\limits_{n \to \infty } \left\| {x_n } \right\|/n = inf\left\{ {\left\| {Tx - x} \right\|:x \in X} \right\} a.e.$$ If, in addition, the spaceX is strictly convex, x n /n converges weakly; and if the norm ofX * is Frechet differentiable (away from zero), x n /n converges strongly.