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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