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Stochastic approximation of Banach-valued random variables with smooth distributions

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Abstract

A random variablef taking values in a Banach spaceE is estimated from another Banach-valued variableg. The best (with respect to theL p-metrix) estimator is proved to exist in the case of Bochnerp-integrable variables. For a Hilbert spaceE andp=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 ofE. 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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Authors and Affiliations

  1. Steklov Mathematical Institute, USSR

    M. O. Smolyanova

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  1. M. O. Smolyanova
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Additional information

Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 425–444, September, 1995.

The author is grateful to B. S. Kashin for posing the problem and helping to write the article, and to the reviewer for a number of useful remarks.

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Smolyanova, M.O. Stochastic approximation of Banach-valued random variables with smooth distributions. Math Notes 58, 970–982 (1995). https://doi.org/10.1007/BF02304775

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  • DOI: https://doi.org/10.1007/BF02304775

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