[Paper Review] A Central Limit Theorem for the Poisson-Voronoi Approximation

Matthias Schulte|arXiv (Cornell University)|Nov 28, 2011

Point processes and geometric inequalities8 references4 citations

TL;DR

This paper establishes a central limit theorem for the volume of the Poisson-Voronoi approximation of a compact convex set $K \subset \mathbb{R}^d$, showing that as the intensity $\lambda$ of the underlying Poisson point process tends to infinity, the normalized volume deviation converges in distribution to a standard normal random variable. The proof relies on Wiener-Itô chaos expansions and an abstract central limit theorem for Poisson functionals, with explicit variance bounds of order $\lambda^{-1 - 1/d}$.

ABSTRACT

For a compact convex set $K$ and a Poisson point process $η$, the union of all Voronoi cells with a nucleus in $K$ is the Poisson-Voronoi approximation of $K$. Lower and upper bounds for the variance and a central limit theorem for the volume of the Poisson-Voronoi approximation are shown. The proofs make use of so called Wiener-Itô chaos expansions and the central limit theorem is based on a more abstract central limit theorem for Poisson functionals, which is also derived.

Motivation & Objective

• To establish a central limit theorem for the volume of the Poisson-Voronoi approximation of a compact convex set $K \subset \mathbb{R}^d$.
• To derive sharp lower and upper bounds for the variance of the approximation volume as a function of the intensity $\lambda$.
• To develop and apply an abstract central limit theorem for Poisson functionals based on Wiener-Itô chaos expansions.
• To extend the results beyond convex sets by identifying weaker geometric conditions under which the central limit theorem still holds.
• To provide quantitative asymptotic bounds on the variance that match the known scaling behavior of $\lambda^{-1 - 1/d}$.

Proposed method

• Utilizes Wiener-Itô chaos expansions to represent the volume of the Poisson-Voronoi approximation as a series of multiple stochastic integrals.
• Applies an abstract central limit theorem for Poisson functionals, derived via Stein's method and Malliavin calculus techniques.
• Employs the coarea formula and geometric estimates to control the $L^2$-norms of the chaos expansion kernels $f_n$.
• Establishes uniform convergence properties of $n! \|f_n\|_n^2$ to verify the conditions of the abstract CLT.
• Uses the Steiner formula and parallel set volume estimates for convex sets to bound the variance terms.
• Derives explicit variance bounds by combining geometric intrinsic volumes $V_i(K)$ with scaling in $\lambda$ and dimension $d$.

Experimental results

Research questions

• RQ1 Does the volume of the Poisson-Voronoi approximation of a convex set $K$ satisfy a central limit theorem as the intensity $\lambda \to \infty$?
• RQ2 What is the precise asymptotic scaling of the variance of the Poisson-Voronoi approximation volume in terms of $\lambda$ and the geometry of $K$?
• RQ3 Can the central limit theorem be extended to non-convex sets under weaker geometric assumptions?
• RQ4 How do the intrinsic volumes $V_i(K)$ influence the variance of the approximation volume?
• RQ5 What is the role of the inradius $r_K$ in determining the asymptotic behavior of the variance?

Key findings

• The normalized volume deviation $\frac{\operatorname{PV}(K) - \operatorname{Vol}(K)}{\sqrt{\operatorname{Var}\,\operatorname{PV}(K)}}$ converges in distribution to a standard normal random variable as $\lambda \to \infty$, establishing a central limit theorem.
• The variance of the Poisson-Voronoi approximation scales asymptotically as $\lambda^{-1 - 1/d}$, matching the known order from prior work.
• A new lower bound for the variance is derived, showing $\operatorname{Var}\operatorname{PV}(K) \geq \underline{C} \kappa_1 V_{d-1}(K) \lambda^{-1 - 1/d}$ for $\lambda \geq (2/r_K)^d$, with $\underline{C} > 0$ depending only on $d$.
• An upper bound is established: $\operatorname{Var}\operatorname{PV}(K) \leq \overline{C} \sum_{i=0}^{d-1} \kappa_{d-i} V_i(K) \lambda^{-2 + i/d}$, which matches the known scaling and improves upon previous results.
• The abstract central limit theorem for Poisson functionals is derived and applied to the volume functional, providing a general tool for future applications.
• The results extend to non-convex sets satisfying (S1) and (S2): bounded parallel set volumes and a positive lower density condition on the outer $r$-neighborhood, respectively.