Weak laws of large numbers in geometric probability

• 成果类型: Article
• 署名作者: Penrose, MD; Yukich, JE
• 署名单位: Durham University; Lehigh University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2003
• 页码: 277-303
• 关键词: minimal spanning-trees; random points; Computational geometry; graphs; asymptotics; limit

摘要

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers for functionals of marked point processes, including statistics of Boolean models.