proceedings of the american mathematical society Volume 100, Number 4, August 1987 ASYMPTOTIC BEHAVIOR OF p-PREDICTIONS FOR VECTOR VALUED RANDOM VARIABLES JUAN A. CUESTA AND CARLOS MATRÁN ABSTRACT. 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]). 1. Introduction. Throughout this paper (fi, a, p) denotes a probability space, (B, || ||) is a uniformly convex Banach space, and Lp(a) = Lp(fl,a,p,B), 1 < p < oo, represents the abstract Lebesgue-Bochner Lp-space. If a is a sub-cr-algebra of er, Lp(a) denotes the (closed) subspace of Lp(a) consisting in all the equivalence classes in Lp(a) containing an a-measurable function. In this notation we will not make any distinction between a random variable and the equivalence class it represents. Recall that random variables in Lp(a) are strongly a-measurable, i.e. they are a.s. limits of finite valued a-measurable random variables. In [1] Ando and Amemiya have introduced the p-predictions given a cr-algebra for real valued random variables. In an analogous way, taking into account that Lp(a), 1 < p < oo, is uniformly convex, we may consider the p-prediction of a variable X E Lp(o) given the sub-cr-algebra a as the (unique) best Lp-approximation to X by elements of Lp(a). Therefore the p-prediction will be continuous in Lp(a). However there exist important differences between the real and the abstract cases. For instance, it is well known that if B = R and p — 2, the 2-prediction given a coincides with the conditional mean given a, while this is not true for general uniformly convex spaces. In fact the conditional mean is always linear (see Diestel and Uhl [8, p. 122]) but the 2-prediction is not linear unless B is a Hubert space or fi is the union of two p>atoms (Herrndorf [10]). This paper deals with the study of the limit of p-predictions as p —* oo. In the remainder of this work we consider a fixed p>essentially bounded random variable X (i.e. X E Loo(a)) and we will prove that the p-prediction of X given the (fixed) sub-cr-algebra a converges to a best Z-oo-approximation of X by elements of Loc(a) (or oo-prediction of X given a). For real valued random variables this result was proved in [3]. Notice that the study of the convergence as p —*oo of p-predictions on uniformly convex spaces Received by the editors June 1, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 60G30, 28B05, 60B12. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 716