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

TL;DR

This paper investigates the asymptotic behavior of the typical Poisson-Voronoi and Crofton cells in the plane as their inradius grows large, establishing laws of large numbers, central limit theorems, and deviation results.

Contribution

It provides new probabilistic limit theorems for large inradius Poisson-Voronoi and Crofton cells, connecting tessellations, convex hulls, and germ-grain models.

Findings

Law of large numbers for vertices and area outside the disk

Central limit theorem for the area of the cell outside the disk

Moderate deviation results for the cell area

Abstract

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.