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June, 1971 The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures
Thomas Kailath
Ann. Math. Statist. 42(3): 1054-1067 (June, 1971). DOI: 10.1214/aoms/1177693332


The Radon-Nikodym derivative (RND) with respect to Wiener measure of a measure determined by the sum of a differentiable (random) signal process and a Wiener process is shown, under rather general conditions, to have the same form as the RND for the case of a known (nonrandom) signal plus a Wiener process. The role of the known signal is played by the causal least-squares estimate of the signal process given the sum process. This formula can be shown to be equivalent to all previously known explicit formulas for RND's relative to Wiener measure. Moreover, and more important, the formula suggests a general structure for engineering approximation and implementation of signal detection schemes. Secondly, an explicit necessary and sufficient characterization, in signal plus noise form, is given of all processes absolutely continuous with respect to a Wiener process. Finally, the results are extended to some reference measures related to Wiener measure, in particular to measures induced by martingales of a Wiener process. We also note that the case where both measures are Gaussian permits some stronger results. The proofs are based on several recent results in martingale theory.


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Thomas Kailath. "The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures." Ann. Math. Statist. 42 (3) 1054 - 1067, June, 1971.


Published: June, 1971
First available in Project Euclid: 27 April 2007

zbMATH: 0246.60041
MathSciNet: MR281279
Digital Object Identifier: 10.1214/aoms/1177693332

Rights: Copyright © 1971 Institute of Mathematical Statistics

Vol.42 • No. 3 • June, 1971
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