Poisson-Delaunay approximation

Abstract

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

Figures (2)

• Figure 1: Poisson-Delaunay approximation of an ellipse.
• Figure 2: Symmetric difference of the Poisson-Delaunay approximation.

Theorems & Definitions (11)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4