Stochastic approximation of Banach-valued random variables with smooth distributions

A random variable $f$ taking values in a Banach space $E$ is estimated from another Banach-valued variable $g$. The best (with respect to the $L_p$-metrix) estimator is proved to exist in the case of Bochner $p$-integrable variables. For a Hilbert space $E$ and $p=2$, the best estimator is expressed in terms of the conditional expectation and, in the case of jointly Gaussian variables, in terms of the orthoprojection on a certain subspace of $E$. More explicit expressions in terms of surface measures are given for the case in which the underlying probability space is a Hilbert space with a smooth probability measure. The results are applied to the Wiener process to improve earlier estimates given by K. Ritter [4].