AI 中文总结
研究一般空间上泊松过程的停止集，利用其空间马尔可夫性质推导分布恒等式，结果可推广到有限泊松过程凸包及更一般随机集的相关恒等式。
AI 中文摘要
我们考虑一般空间上强度测度为λ的泊松过程η，以及依赖于η的停止集Z。特别利用η的空间马尔可夫性质，我们推导了η与λ在Z及Z补集上限制的若干分布恒等式。欧氏空间中的一个重要特例是有限泊松过程的凸包，此情形下我们的结果推广了连接凸包顶点数与体积的经典及最新恒等式。我们的结果适用于一般泊松包络，且主要甚至适用于既非有界也非停止集的更一般随机集。
英文摘要
We consider a Poisson process $η$ on a general space with intensity measure $λ$ and a stopping set $Z$ depending on $η$. Using in particular the spatial Markov property of $η$, we derive several distributional identities for the restrictions of $η$ and $λ$ to $Z$ and the complement of $Z$. An important special case in Euclidean space is the convex hull of a finite Poisson process. In this case our results generalize classical (and also more recent) identities connecting the number of vertices and the volume of the convex hull. Our results apply to general Poisson hulls and predominantly even to more general random sets which are neither assumed to be bounded nor to be stopping sets.
Comments25 pages
