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

Elon Kohlberg*, Abraham Neyman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

A map T: X → X on a normed linear space is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ ∀x,y ∈ X. Let (Ω, Σ, P) be a probability space, ℱ0 ⊂ ℱ1 ⊂ ⋯ ⊂ ℱn an increasing chain of σ-fields spanning Σ, X a Banach space, and T: X → X. A sequence (xn) of strongly ℱn-measurable and strongly P-integrable functions on Ω taking on values in X is called a T-martingale if Ε(xn+1 | ℱn) = T(xn). Let T: H → H be a nonexpansive mapping on a Hubert space H, and let (xn) be a T-martingale taking on values in H. If Σ n=1 n-2 E∥xn+1 - Txn2 < ∞ then xn/n converges a.e. Let T: X → X be a nonexpansive mapping on a p-uniformly smooth Banach space X, 1 < p ≤ 2, and let (xn) be a T-martingale (taking on values in X). If Σ n-p E (∥xn - Txn-1p) < ∞ then there exists a continuous linear functional f ∈ X* of norm 1 such that lim n→∞ f(xn)/n= lim n→∞ ∥xn∥/n = inf {∥ Tx - x ∥ : x ∈X} a.e. If, in addition, the space X is strictly convex, xn/n converges weakly; and if the norm of X* is Fréchet differentiable (away from zero), xn/n converges strongly.

Original languageEnglish
Pages (from-to)93-108
Number of pages16
JournalIsrael Journal of Mathematics
Volume111
DOIs
StatePublished - 1999

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