关于随机场水平集测度的矩的有限性
On the finiteness of the moments of the measure of level sets of random fields
摘要：
给出了光滑实值随机场水平集测度矩有限的一般条件。作为副积，当二阶矩一致有界时，得到了这些水平集的测度期望的一个新的广义Kac-Rice公式（KRF）。这些条件包括：（i）轨道的可微性达到一定的顺序k，（ii）的有限性的时刻的第k次偏导数的字段到另一个订单，（iii）的联合密度的字段和它的一些衍生物的有界性。特别注意散粒噪声的过程和领域。其他应用包括平稳高斯过程，卡方过程和正则化扩散过程。AMS 2000分类：Primary 60 G60。第60 G15
英文摘要：
General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.
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