×

Stochastic approximation of Banach-valued random variables with smooth distributions. (English. Russian original) Zbl 0851.60003

Math. Notes 58, No. 3, 970-982 (1995); translation from Mat. Zametki 58, No. 3, 425-444 (1995).
Summary: 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 \(L_p\)-metric) 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 [J. Complexity 6, No. 4, 337-364 (1990; Zbl 0718.41046)].

MSC:

60B05 Probability measures on topological spaces

Citations:

Zbl 0718.41046
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Loève,Probability Theory, D. Van Nostrand Company Inc., Princeton (1960).
[2] I. I. Gikhman and A. V. Skorokhod,Theory of Random Processes [in Russian], Vol. 1, Nauka, Moscow (1970). · Zbl 0132.37902
[3] J. F. Traub, G. W. Wasilkowski, and H. Woźniakowski,Information-Based Complexity, Academic Press, New York (1988).
[4] K. Ritter, ”Approximation and optimization on the Wiener space,”J. Complexity,6, 337–364 (1990). · Zbl 0718.41046 · doi:10.1016/0885-064X(90)90027-B
[5] E. Novak and K. Ritter,Some Complexity Results for Zero Finding for Univariate Functions, Preprint (1992). · Zbl 0771.65024
[6] D. Lee, ”Approximation of linear operators on a Wiener space,”Rocky Mount. J. Math.,16, 641–659 (1986). · Zbl 0679.41014 · doi:10.1216/RMJ-1986-16-4-641
[7] A. V. Skorokhod,Integration on Hilbert Space [in Russian], Nauka, Moscow (1975). · Zbl 0355.62084
[8] A. V. Uglanov, ”Surface measures in Banach spaces,”Mat. Sb. [Math. USSR-Sb.],110, No. 4, 189–217 (1979).
[9] N. Bourbaki,Éléments de mathématique. Livre VI Intégration, Hermann, Paris (1965).
[10] J. Diestel and O. F. Uhl,Vector Measures, American Mathematical Society, Providence, R. I. (1977).
[11] J. Neveu,Bases mathématiques du calcul des probabilités, Masson et Cie, Paris (1964). · Zbl 0137.11203
[12] V. I. Bogachev and O. G. Smolyanov, ”Analytical properties of infinite-dimensional distributions,”Uspekhi Mat. Nauk [Russian Math. Surveys],45, No. 3, 1–83 (1990).
[13] M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York (1975). · Zbl 0308.47002
[14] H. -S. Kuo,Gaussian Measures in Banach Spaces, Springer-Verlag, Berlin-Heidelberg-New York (1975). · Zbl 0306.28010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.