ASYMPTOTIC BEHAVIOR OF p-PREDICTIONS FOR VECTOR VALUED RANDOM VARIABLES

Let (Q,<r, p) be a probability space and let X be a B-valued /i-essentially bounded random variable, where (B,\\ ||) is a uniformly convex Banach space. Given a, a sub-<r-algebra of a, the p-prediction (1 < p < oo) of X is defined as the best Lp-approximation to X by (»-measurable random variables. The paper proves that the Pólya algorithm is successful, i.e. the p-prediction converges to an "co-prediction" as p —» oo. First the proof is given for p-means (p-predictions given the trivial ir-algebra), and the general case follows from the characterization of the p-prediction in terms of the p-mean of the identity in B with respect to a regular conditional probability. Notice that the problem was treated in [7], but the proof is not satisfactory (as pointed out in [4]).