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

Cramér and Leadbetter introduced a sufficient condition to have a finite variance of the zeros number 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 taht 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 conditiion.

KRATZ, M. and LEON, J. (2006). On the second moment of the number of crossings by a stationary Gaussian process. Annals of Probability, 34(4), pp. 1601-1607.

Mots clés : #AMS-classification, #60G15, #60G10, #60G70Crossings, #Gaussian-processes, #Geman-condition, #Hermite-polynomials, #level-curve, #spectral-moment