Poisson-Laguerre 剖分的大内切半径的点过程收敛性

中文摘要 AI 辅助

本文研究了 Poisson-Voronoi 剖分的一种加权推广，称为 Poisson-Laguerre 剖分，其中生成 Poisson 过程的核额外携带独立的非负随机权重。对于每个胞腔，我们可将其内切半径定义为以核为中心且包含于该胞腔内的最大球的半径。我们考虑核、权重和内切半径的点过程，其中核取自不断增大的观测窗口，并对过程进行适当的缩放和平移以观察大内切半径的行为。我们证明了其依分布收敛于适当的 Poisson 过程，并由此得到最大内切半径的渐近行为作为推论。我们的结果涵盖维度 $d \geq 3$ 且有界随机权重、维度 $d = 2$ 且随机权重具有适当的有限指数矩，以及维度 $d \geq 2$ 且随机权重服从幂律分布的情形。特别地，我们观察到在权重有界时，Poisson-Laguerre 剖分在平面情形与高维情形下表现出不同的行为。我们结果的证明基于适当的 Poisson 过程逼近技术以及对大胞腔几何的细致研究。

英文摘要

In this paper we study a weighted generalization of the Poisson-Voronoi tessellation called the Poisson-Laguerre tessellation, where the nuclei of the generating Poisson process additionally carry independent non-negative random weights. For each cell we can define its inradius as the radius of the largest ball centered at the nucleus and contained in the cell. We consider point processes of nuclei, weights and inradii, where the nuclei are taken from growing observation windows and the processes are suitably rescaled and shifted to see the behavior of large inradii. We prove convergence in distribution to suitable Poisson processes, obtaining as corollaries the asymptotic behavior of the maximal inradii. Our results cover dimension $d \geq 3$ and bounded random weights, dimension $d = 2$ and random weights with suitable finite exponential moments as well as dimension $d \geq 2$ and random weights following a power-law distribution. Particularly, we observe a different behavior of the Poisson-Laguerre tessellation in the planar and higher dimensional cases when the weights are bounded. The proofs of our results are based on suitable Poisson process approximation techniques and a careful investigation of the geometry of large cells.

发表机构

• Hamburg University of Technology（汉堡工业大学）
• Charles University（查理大学）

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