A strong law of large numbers for nonexpansive vector-valued stochastic processes期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Elon Kohlberg Abraham Neyman

Affiliation:	(1) Graduate School of Business Administration, Harvard University, Soldiers Field Road, 02163 Boston, MA, USA;(2) Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel;(3) The Institute for Decision Sciences, SUNY at Stony Brook, 11794-4384 Stony Brook, NY, USA

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 (x_n) 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 (x_n) be aT-martingale taking on values inH. If $sumlimits_{n = 1}^infty {n^{ - 2} } Eleft\| {x_{n + 1} - Tx_n } right\|^2< infty ,$ then x_n/n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1n) be aT-martingale (taking on values inX). If $sum n ^{ - p} Eleft( {left\| {x_n - Tx_{n - 1} } right\|^p } right)< infty ,$ then there exists a continuous linear functionalf∈X ^* of norm 1 such that $mathop {lim }limits_{n to infty } f(x_n )/n = mathop {lim }limits_{n to infty } left\| {x_n } right\|/n = infleft{ {left\| {Tx - x} right\|:x in X} right} a.e.$ If, in addition, the spaceX is strictly convex, x_n/n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x_n/n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093

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