Linear estimators and measurable linear transformations on a Hilbert space
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We consider the problem of estimating the mean of a Gaussian random vector with values in a Hilbert space. We argue that the natural class of linear estimators for the mean is the class of measurable linear transformations. We give a simple description of all measurable linear transformations with respect to a Gaussian measure. If X and θ are jointly Gaussian then E[θ¦X] is a measurable linear transformation. As an application of the general theory we describe all measurable linear transformations with respect to the Wiener measure in terms of Wiener integrals.
KeywordsHilbert Space Conditional Expectation Gaussian Measure Trace Class Continuous Linear Operator
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- 3.Kuo, H.: Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics 403. BerlinHeidelberg-New York: Springer 1975Google Scholar
- 6.Lipster, R.S., Shiryayev, A.N.: Statistics of Random Processes I, General Theory. Berlin-Heidelberg-New York: Springer 1977Google Scholar
- 7.Reed, M., Simon, B.: Functional Analysis I (Revised and Enlarged Edition). New York: Academic Press 1980Google Scholar
- 8.Rozanov, J.A.: Infinite-dimensional Gaussian distribution. American Mathematical Society (Trans. Proceedings of the Steklov Institute of Mathematics 108) (1971)Google Scholar
- 10.Šilov, G.E., Fan Dyk Tin: Integral, Measure and Derivative on Linear Spaces. In Russian. Moscow: Nauka 1967Google Scholar