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{1158673329,
author = {Kratz, Marie F. and Le\'on, Jos\'e R.},
title = {On the second moment of the number of crossings by a stationary Gaussian process},
journal = {Ann. Probab.},
volume = {34},
number = {1},
year = {2006},
pages = { 1601-1607},
language = {en},
url = {http://dml.mathdoc.fr/item/1158673329}
}
Kratz, Marie F.; León, José R. On the second moment of the number of crossings by a stationary Gaussian process. Ann. Probab., Tome 34 (2006) no. 1, pp. 1601-1607. http://gdmltest.u-ga.fr/item/1158673329/