论文标题
泊松超平面相交点过程的弱收敛性
Weak convergence of the intersection point process of Poisson hyperplanes
论文作者
论文摘要
本文介绍了$ \ mathbb {r}^d $强度$ t> 0 $中的固定和各向同性泊松超平面过程的交点过程，其中只考虑了与半径$ r> 0 $相交的超级平面。服用$ r = t^{ - \ frac {d} {d+1}} $，这表明，此点过程在分布中以$ t \ to \ infty $的形式收敛到$ \ mathbb {r}^r}^d \ setMinus \ setMinus \ {0 \}的强度量的Poisson Point Postes $ \ | x \ |^{ - （D+1）} $相对于Lebesgue度量。还提供了与坎托维奇 - 罗宾斯坦距离有关的收敛速度的结合。背景是抽象泊松空间上的一般功能泊松近似定理。还讨论了相交点过程的凸壳的弱收敛性以及其$ f $ - 矢量的收敛性的影响，从而在计算几何学中讨论了devroye和devroye和toussaint的猜想[J. \算法14.3（1993），381--394]。
This paper deals with the intersection point process of a stationary and isotropic Poisson hyperplane process in $\mathbb{R}^d$ of intensity $t>0$, where only hyperplanes that intersect a centred ball of radius $R>0$ are considered. Taking $R=t^{-\frac{d}{d+1}}$ it is shown that this point process converges in distribution, as $t\to\infty$, to a Poisson point process on $\mathbb{R}^d\setminus\{0\}$ whose intensity measure has power-law density proportional to $\|x\|^{-(d+1)}$ with respect to the Lebesgue measure. A bound on the speed of convergence in terms of the Kantorovich-Rubinstein distance is provided as well. In the background is a general functional Poisson approximation theorem on abstract Poisson spaces. Implications on the weak convergence of the convex hull of the intersection point process and the convergence of its $f$-vector are also discussed, disproving and correcting thereby a conjecture of Devroye and Toussaint [J.\ Algorithms 14.3 (1993), 381--394] in computational geometry.
