Limit theorems for the typical Poisson-Voronoi cell and the Crofton cell with a large inradius

Cluster size distributions of extreme values for the Poisson-Voronoi tessellation

We consider the Voronoi tessellation based on a homogeneous Poisson point process in $\mathbf{R}^{d}$. For a geometric characteristic of the cells (e.g. the inradius, the circumradius, the volume), we investigate the point process of the nuclei of the cells with large values. Conditions are obtained for the convergence in distribution of this point process of exceedances to a homogeneous compound Poisson point process. We provide a characterization of the asymptotic cluster size distribution which is based on the Palm version of the point process of exceedances. This characterization allows us…

Extreme values for characteristic radii of a Poisson-Voronoi tessellation

A homogeneous Poisson-Voronoi tessellation of intensity $γ$ is observed in a convex body $W$. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in $W$. We prove that when $γ\rightarrow\infty$, these four quantities converge to Gumbel or Weibull distributions up to a rescaling…

高维中典型泊松-沃罗诺伊胞腔的形状

On the shape of the typical Poisson-Voronoi cell in high dimensions

本文研究高维泊松-沃罗诺伊镶嵌典型胞腔的几何形状，证明其内切半径、外接半径、直径和平均宽度经适当标度后依概率收敛到确定常数，并刻画了低维面直径远小于整体胞腔直径以及面数量的渐近公式。

A general study of extremes of stationary tessellations with applications

Let $\mathfrak{m}$ be a random tessellation in $\mathbf{R}^d$ observed in a bounded Borel subset $W$ and $f(\cdot)$ be a measurable function defined on the set of convex bodies. To each cell $C$ of $\mathfrak{m}$ we associate a point $z(C)$ which is the nucleus of $C$. Applying $f(\cdot)$ to all the cells of $\mathfrak{m}$, we investigate the order statistics of $f(C)$ over all cells $C\in\mathfrak{m}$ with nucleus in $\mathbf{W}_ρ=ρ^{1/d}W$ when $ρ$ goes to infinity. Under a strong mixing property and a local condition on $\mathfrak{m}$ and $f(\cdot)$, we show a general theorem which reduces …

Poisson-Laguerre 剖分的大内切半径的点过程收敛性

Point process convergence of large inradii of Poisson-Laguerre tessellations

本文研究 Poisson-Laguerre 剖分中随观测窗口增大而缩放的大内切半径点过程，证明其依分布收敛于 Poisson 过程，并给出最大内切半径的渐近行为，覆盖多种维度与权重分布。