Poisson-Delaunay approximation

Abstract

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