On the second moment of the number of crossings by a stationary Gaussian process
arXiv:math/0609682 · doi:10.1214/009117906000000142
Abstract
Cramér and Leadbetter introduced in 1967 the sufficient condition \[\frac{r''(s)-r''(0)}{s}\in L^1([0,δ],dx),\qquad δ>0,\] to have a finite variance of the number of zeros of a centered stationary Gaussian process with twice differentiable covariance function . 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.
Published at http://dx.doi.org/10.1214/009117906000000142 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)