The Annals of Probability

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

Marie F. Kratz and José R. León

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Abstract

Cramér and Leadbetter introduced in 1967 the sufficient condition

\[\frac{r''(s)-r''(0)}{s}\in L^{1}([0,\delta],dx),\qquad \delta>0,\]

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.

Article information

Source
Ann. Probab., Volume 34, Number 4 (2006), 1601-1607.

Dates
First available in Project Euclid: 19 September 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1158673329

Digital Object Identifier
doi:10.1214/009117906000000142

Mathematical Reviews number (MathSciNet)
MR2257657

Zentralblatt MATH identifier
1101.60024

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G10: Stationary processes 60G70: Extreme value theory; extremal processes

Keywords
Crossings Gaussian processes Geman condition Hermite polynomials level curve spectral moment

Citation

Kratz, Marie F.; León, José R. On the second moment of the number of crossings by a stationary Gaussian process. Ann. Probab. 34 (2006), no. 4, 1601--1607. doi:10.1214/009117906000000142. https://projecteuclid.org/euclid.aop/1158673329


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References

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