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June, 1972 On the Variance of the Number of Zeros of a Stationary Gaussian Process
Donald Geman
Ann. Math. Statist. 43(3): 977-982 (June, 1972). DOI: 10.1214/aoms/1177692560

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.

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Donald Geman. "On the Variance of the Number of Zeros of a Stationary Gaussian Process." Ann. Math. Statist. 43 (3) 977 - 982, June, 1972. https://doi.org/10.1214/aoms/1177692560

Information

Published: June, 1972
First available in Project Euclid: 27 April 2007

zbMATH: 0244.60029
MathSciNet: MR301791
Digital Object Identifier: 10.1214/aoms/1177692560

Rights: Copyright © 1972 Institute of Mathematical Statistics

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Vol.43 • No. 3 • June, 1972
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