Asymptotic behavior of $p$-predictions for vector valued random variables

Abstract:

Let $(\Omega ,\sigma ,\mu )$ be a probability space and let $X$ be a $B$-valued $\mu$-essentially bounded random variable, where $(B,\left \| {} \right \|)$ is a uniformly convex Banach space. Given $\alpha$, a sub-$\sigma$-algebra of $\sigma$, the $p$-prediction $1 < p < \infty$ of $X$ is defined as the best ${L_p}$-approximation to $X$ by $\alpha$-measurable random variables. The paper proves that the Pólya algorithm is successful, i.e. the $p$-prediction converges to an "$\infty$-prediction" as $p \to \infty$. First the proof is given for $p$-means ($p$-predictions given the trivial $\sigma$-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]).