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

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.