The variance of the number of zeros of a stationary normal process
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Let x(t) be a separable stationary normal stochastic process with zero mean, covariance function r(r) (with r(0) = 1, for convenience), and spectrum ^(X) having an absolutely continuous component. Let N denote the number of times x(t) crosses the zero level in Q^t^T and write X2 for the second spectral moment JQ\ dF(X) = —r"(0). Then it is well known that the mean of the random variable N is given by T V ^ A * (and, in fact, it has been recently shown by Ylvisaker [4] that this is true whether or not X2 is finite, provided x(t) has continuous sample functions with probability one). The second moment has been given by a number of authors (e.g., [2], [3]) but the best conditions available to date include the existence of a sixth derivative for the covariance function r(r). Here we give the following result for the second moment of N and indicate briefly the general lines of proof. Full details will appear elsewhere.
[1] P. Schultheiss,et al. Short‐Time Frequency Measurement of Narrow‐Band Random Signals by Means of a Zero Counting Process , 1955 .
[2] V. Volkonskii,et al. Some Limit Theorems for Random Functions. II , 1959 .
[3] M. R. Leadbetter. On crossings of arbitrary curves by certain Gaussian processes , 1965 .