A convexity condition in Banach spaces and the strong law of large numbers

In a recent paper [l], this author showed this theorem under the hypotheses that ï is uniformly convex and that the variances of Xi are uniformly bounded (Var(Xi) =E(||Xi||2)). At the same time, an example was given of a space in which the theorem fails. It is now possible, using the methods of [l], to show a necessary and sufficient condition on the Banach space ï to yield this particular strong law of large numbers. A Banach space 3É is said to have property (A) if, for every sequence {Xi} oí independent random X-variables with E(Xi)=0, all i, and Var(Xi) <M, all i, we have