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 Fréchet differentiable (away from zero), x n /n converges strongly.
Similar content being viewed by others
References
Y. S. Chow,Local convergence of martingales and the law of large numbers, Annals of Mathematical Statistics,36 (1965), 552–558.
J. Diestel,Geometry of Banach Spaces—Selected Topics, Springer-Verlag, Berlin, 1975.
W. Feller,An Introduction to Probability Theory, Vol. II, Wiley, New York, 1965.
J. Hoffmann-Jorgensen and G. Pisier,The law of large numbers and the Central Limit Theorem in Banach spaces, Annals of Probability4 (1976), 587–599.
E. Kohlberg and A. Neyman,Asymptotic behavior of nonexpansive mappings in normed linear spaces, Israel Journal of Mathematics38 (1981), 269–275.
W. A. Woyczhyńki,Laws of large numbers for vector-valued martingales. Bulletin de l’Académai Polonaise des Sciences23 (1975), 1199–1201.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Science Foundation Grant MCS-82-02093
Rights and permissions
About this article
Cite this article
Kohlberg, E., Neyman, A. A strong law of large numbers for nonexpansive vector-valued stochastic processes. Isr. J. Math. 111, 93–108 (1999). https://doi.org/10.1007/BF02810679
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02810679