Geodesic
Stochastic approximation of Banach-valued random variables with smooth distributions
Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 425-444 
M. O. Smolyanova. Stochastic approximation of Banach-valued random variables with smooth distributions. Matematičeskie zametki, Tome 58 (1995) no. 3, pp. 425-444. http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a10/
@article{MZM_1995_58_3_a10,
     author = {M. O. Smolyanova},
     title = {Stochastic approximation of {Banach-valued} random variables with smooth distributions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {425--444},
     year = {1995},
     volume = {58},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_3_a10/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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].

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