On the convergence of certain sums of independent random elements

Our notation is standard ([1], [3], [4], [9]). Throughout this note ∆ will denote the Cantor space {−1, 1}, Σ the σ-algebra of subsets of ∆ generated by the n-cylinders of ∆ for each n ∈ N, and ν the Borel probability ⊗i=1νi on Σ, where νi : 2 {−1,1} → [0, 1] is defined by νi(∅) = 0, νi({−1}) = νi({1}) = 1/2 and νi({−1, 1}) = 1 for each i ∈ N. In what follows X will be a real Banach space and L0(ν,X) will stand for the (F )-space over R of all [classes of] ν-measurable X-valued functions equipped with the (F )-norm