The Annals of Mathematical Statistics

The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures

Thomas Kailath

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Abstract

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.

Article information

Source
Ann. Math. Statist., Volume 42, Number 3 (1971), 1054-1067.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177693332

Digital Object Identifier
doi:10.1214/aoms/1177693332

Mathematical Reviews number (MathSciNet)
MR281279

Zentralblatt MATH identifier
0246.60041

JSTOR
links.jstor.org

Citation

Kailath, Thomas. The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures. Ann. Math. Statist. 42 (1971), no. 3, 1054--1067. doi:10.1214/aoms/1177693332. https://projecteuclid.org/euclid.aoms/1177693332


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