## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 3 (1972), 977-982.

### On the Variance of the Number of Zeros of a Stationary Gaussian Process

#### Abstract

For a real, stationary Gaussian process $X(t)$, it is well known that the mean number of zeros of $X(t)$ in a bounded interval is finite exactly when the covariance function $r(t)$ is twice differentiable. Cramer and Leadbetter have shown that the variance of the number of zeros of $X(t)$ in a bounded interval is finite if $(r"(t) - r"(0))/t$ is integrable around the origin. We show that this condition is also necessary. Applying this result, we then answer the question raised by several authors regarding the connection, if any, between the existence of the variance and the existence of continuously differentiable sample paths. We exhibit counterexamples in both directions.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 3 (1972), 977-982.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692560

**Digital Object Identifier**

doi:10.1214/aoms/1177692560

**Mathematical Reviews number (MathSciNet)**

MR301791

**Zentralblatt MATH identifier**

0244.60029

**JSTOR**

links.jstor.org

#### Citation

Geman, Donald. On the Variance of the Number of Zeros of a Stationary Gaussian Process. Ann. Math. Statist. 43 (1972), no. 3, 977--982. doi:10.1214/aoms/1177692560. https://projecteuclid.org/euclid.aoms/1177692560