Asymptotic properties of random Voronoi cells with arbitrary underlying density. (June 2020)

• Record Type: Journal Article
• Authors: Gibbs, Isaac; Chen, Linan
• Is Part Of: Advances in applied probability. Volume 52:Number 2(2020)
• Page Start: 655
• Page End: 680
• Publication Date: 2020-06
• Subjects: Point process with arbitrary density, -- geometry of random Voronoi diagram, -- convergence in distribution; 60G55, -- 60D05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.12
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: We consider the Voronoi diagram generated by n independent and identically distributed $\mathbb{R}^{d}$ -valued random variables with an arbitrary underlying probability density function f on $\mathbb{R}^{d}$, and analyze the asymptotic behaviors of certain geometric properties, such as the measure, of the Voronoi cells as n tends to infinity. We adapt the methods used by Devroye et al. (2017) to conduct a study of the asymptotic properties of two types of Voronoi cells: (1) Voronoi cells that have a fixed nucleus; (2) Voronoi cells that contain a fixed point. We show that the geometric properties of both types of cells resemble those in the case when the Voronoi diagram is generated by a homogeneous Poisson point process. Additionally, for the second type of Voronoi cells, we determine the limiting distribution, which is universal in all choices of f, of the re-scaled measure of the cells.