## The Annals of Mathematical Statistics

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

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

#### 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