Poisson-Delaunay approximation

TL;DR

This paper studies the Poisson-Delaunay approximation of a set using a Poisson point process, providing unbiasedness, variance bounds, a CLT, and asymptotic analysis of the approximation error.

Contribution

It introduces the Poisson-Delaunay approximation for sets and establishes its statistical properties, including unbiasedness, variance bounds, and asymptotic behavior.

Findings

$ ext{Vol}_d(A_{ ext{Poisson-Delaunay}})$ is an unbiased estimator for $ ext{Vol}_d(A)$.

Variance bounds and a quantitative central limit theorem are derived.

Asymptotic behavior of the symmetric difference measure is characterized as $t o abla$.

Abstract

For a Borel set $A$ and a stationary Poisson point process $\eta_t$ in $\mathbb R^d$ of intensity $t>0$, the Poisson-Delaunay approximation $ A_{\eta_t}$ of $A$ is the union of all Delaunay cells generated by $\eta_t$ with center in $A$. It is shown that $\lambda_d(A_{\eta_t})$ is an unbiased estimator for $\lambda_d(A)$, variance bounds and a quantitative central limit theorem are given. The asymptotic behaviour of the symmetric difference $\lambda_d(A\Delta A_{\eta_t})$ is derived as $t \to\infty$.