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

Cramer and Leadbetter introduced in 1967 the sufficient condition r"(s)-r"(0)/s is an element of 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.