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On the Variance of the Number of Zeros of a Stationary Gaussian Process
The Annals of Mathematical Statistics
Vol. 43, No. 3 (Jun., 1972), pp. 977-982
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2240391
Page Count: 6
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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.
The Annals of Mathematical Statistics © 1972 Institute of Mathematical Statistics