On the second moment of the number of crossings by a stationary Gaussian process
On the second moment of the number of crossings by a stationary Gaussian process
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DOI：
10.1214/009117906000000142
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发表时间：
2006-07-01
影响因子：
2.3
通讯作者：
Leon, Jose R.
中科院分区：
文献类型：
--
作者：
Kratz, Marie F.;Leon, Jose R.
Cramer和Leadbetter在1967年引入了一个充分条件r”（s）-r”（0）/s是L-1（[0，delta]，dx）中的一个元素，delta > 0，使得协方差函数为二次可微的中心平稳高斯过程的零点个数的方差有限。这个条件被称为Geman条件，因为Geman在1972年证明了它也是一个必要条件。到目前为止，没有这样的标准是已知的计数的水平比其他的平均值的交叉点，本文表明，Geman条件仍然是充分和必要的，有一个有限的方差的任何固定的水平交叉点的数量。为了推广到曲线交叉点的数量，必须将曲线上的条件添加到Geman条件。
Cramer and Leadbetter introduced in 1967 the sufficient conditionr"(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.