Criteria for Poisson process convergence with applications to inhomogeneous Poisson-Voronoi tessellations

TL;DR

This paper establishes conditions for the weak convergence of point processes to a Poisson process, with applications to Bernoulli sequences and inhomogeneous Poisson-Voronoi tessellations, advancing understanding of stochastic spatial models.

Contribution

It introduces new criteria for Poisson process convergence and applies them to inhomogeneous Poisson-Voronoi tessellations and Bernoulli sequences.

Findings

Derived conditions for weak convergence of point processes to Poisson processes.

Applied criteria to inhomogeneous Poisson-Voronoi tessellations.

Analyzed starting points of k-runs in Bernoulli sequences.

Abstract

This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables and point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.