## The Annals of Mathematical Statistics

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

Donald Geman

#### 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

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

#### 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