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

• 成果类型: Article
• 署名作者: Calka, P; Schreiber, T
• 署名单位: Universite Paris Cite; Nicolaus Copernicus University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000134
• 发表日期: 2005
• 页码: 1625-1642
• 关键词: random convex-hull; geometric characteristics; random polytopes; distributions; tessellation; approximation; number; plane; sets

摘要

In this paper, we are interested in the behavior of the typical Poisson-Voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity. We prove a law of large numbers for its number of vertices and the area of the cell outside the disk. Moreover, for the latter, we establish a central limit theorem as well as moderate deviation type results. The proofs deeply rely on precise connections between Poisson-Voronoi tessellations, convex hulls of Poisson samples and germ-grain models in the unit ball. Besides, we derive analogous facts for the Crofton cell of a stationary Poisson line process in the plane.