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

• 成果类型: Article
• 署名作者: Kratz, Marie F.; Leon, Jose R.
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; University of Central Venezuela
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000142
• 发表日期: 2006
• 页码: 1601-1607
• 关键词: zeros

摘要

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.