## The Annals of Probability

- Ann. Probab.
- Volume 1, Number 1 (1973), 131-137.

### Occupation Times for Smooth Stationary Processes

D. Geman and J. Horowitz

#### Abstract

An occupation-time density is identified for a class of absolutely continuous functions $x(t)$ in terms of $x'(t)$ and the number of times that $x(t)$ assumes the values in its range. This result is applied to stationary random processes with a finite second spectral moment. As a by-product, a generalization of Rice's formula for the mean number of crossings is obtained.

#### Article information

**Source**

Ann. Probab. Volume 1, Number 1 (1973), 131-137.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176997029

**Digital Object Identifier**

doi:10.1214/aop/1176997029

**Mathematical Reviews number (MathSciNet)**

MR350833

**Zentralblatt MATH identifier**

0264.60027

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 60G17: Sample path properties 60G15: Gaussian processes

**Keywords**

Occupation-time density stationary process level crossings

#### Citation

Geman, D.; Horowitz, J. Occupation Times for Smooth Stationary Processes. Ann. Probab. 1 (1973), no. 1, 131--137. doi:10.1214/aop/1176997029. https://projecteuclid.org/euclid.aop/1176997029