## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 3 (1984), 922-925.

### Randomly Started Signals with White Noise

Burgess Davis and Itrel Monroe

#### Abstract

It is shown that if $B(t), t \geq 0$, is a Wiener process, $U$ is an independent random variable uniformly distributed on (0, 1), and $\varepsilon$ is a constant, then the distribution of $B(t) + \varepsilon \sqrt{(t - U)^+}, 0 \leq t \leq 1$, is absolutely continuous with respect to Wiener measure on $C\lbrack 0, 1\rbrack$ if $0 < \varepsilon < 2$, and singular with respect to this measure if $\varepsilon > \sqrt 8$.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 3 (1984), 922-925.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993243

**Digital Object Identifier**

doi:10.1214/aop/1176993243

**Mathematical Reviews number (MathSciNet)**

MR744249

**Zentralblatt MATH identifier**

0599.60041

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J65: Brownian motion [See also 58J65]

Secondary: 60G30: Continuity and singularity of induced measures 60G17: Sample path properties

**Keywords**

Brownian motion paths

#### Citation

Davis, Burgess; Monroe, Itrel. Randomly Started Signals with White Noise. Ann. Probab. 12 (1984), no. 3, 922--925. doi:10.1214/aop/1176993243. https://projecteuclid.org/euclid.aop/1176993243