A probability inequality for sums of independent Banach space valued random variables.

Let (B, ∥ · ∥) be a real separable Banach space. Let ϕ(·) and ψ(·) be two continuous and increasing functions defined on [0,∞) such that ϕ(0) = ψ(0) = 0, limt→∞ϕ(t) = ∞, and ψ(·)/ϕ(·) is a nondecreasing function on [0,∞). Let {Vn; n ≥ 1} be a sequence of independent and symmetric B-valued random variables. In this note, we establish a probability inequality for sums of independent B-valued random variables by showing that for every n ≥ 1 and all t ≥ 0, ...where an = ϕ(n) and bn = ψ(n), n ≥ 1. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed B-valued random variables. [ABSTRACT FROM AUTHOR]