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

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