On the second moment of the number of crossings by a stationary Gaussian process.

Abstract : Cramér and Leadbetter introduced in 1967 the sufficient condition [(r''(s)-r''(0))/s ] \in L^1([0,\delta],dx), \delta>0, to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function r. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sufficient and necessary to have a finite variance of the number of any fixed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.
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Submitted on : Monday, October 15, 2007 - 1:43:28 PM
Last modification on : Sunday, January 19, 2020 - 6:38:29 PM

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Marie Kratz, José R. Leon. On the second moment of the number of crossings by a stationary Gaussian process.. Annals of Probability, Institute of Mathematical Statistics, 2006, 34 (4), pp.1601-1607. ⟨10.1214/009117906000000142⟩. ⟨hal-00179359⟩

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