A “coboundary” Theorem for Sums of Random Variables Taking Their Values in a Banach Space

For a given metric space (S, d), let us use the term “standard σ-field” to denote the σ-field S of subsets of S generated by the open balls (in the metric d). A function f : S×S×S× . . .→ S is “measurable” (with respect to S) if for every set A ∈ S one has that f−1(A) is a member of S×S×S×. . . , the product σ-field on S × S × S × . . . . Suppose B is a separable real Banach space. Suppose (Ω,F , P ) is a probability space. A “B-valued random variable” is of course a function X : Ω→ B such that, letting B denote the standard σ-field on B, for every set A ∈ B one has that X−1(A) ∈ F . Given a sequence (Xk, k ∈ Z) of B-valued random variables, for any pair of integers J ≤ L we shall denote the “partial sum”