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October, 1975 Conditions for Finite Moments of the Number of Zero Crossings for Gaussian Processes
Jack Cuzick
Ann. Probab. 3(5): 849-858 (October, 1975). DOI: 10.1214/aop/1176996271


Let $M_k(0, T)$ denote the $k$th (factorial) moment of the number of zero crossings in time $T$ by a stationary Gaussian process. We present a necessary and sufficient condition for $M_k(0, T)$ to be finite. This condition is then applied to processes whose covariance functions $\rho(t)$ satisfy the local condition. $$\rho(t) = 1 - \frac{t^2}{2} + \frac{C|t|^3}{6} + o|t|^3$$ for $t$ near zero $(C > 0)$. In this case we show all the crossing moments $M_k(0, T)$ are finite. In the course of the proof of this result, we point out an error which vitiates the related work of Piterbarg (1968) and Mirosin (1971, 1973, 1974a, 1974b). We also find a counterexample to Piterbarg's results.


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Jack Cuzick. "Conditions for Finite Moments of the Number of Zero Crossings for Gaussian Processes." Ann. Probab. 3 (5) 849 - 858, October, 1975.


Published: October, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0328.60023
MathSciNet: MR388515
Digital Object Identifier: 10.1214/aop/1176996271

Primary: 60G17
Secondary: 60G10 , 60G15

Keywords: factorial moments , Gaussian processes , level crossings , moments , Point processes , Zero crossings

Rights: Copyright © 1975 Institute of Mathematical Statistics


Vol.3 • No. 5 • October, 1975
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