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Gaussian approximation for Extreme Points in Laguerre tessellations

Chinmoy Bhattacharjee, Anna Gusakova

TL;DR

The article analyzes Gaussian approximations for the number of extreme points in Poisson-Laguerre tessellations across $\beta$-, $\beta'$-, and Gaussian-Voronoi models. Using region stabilization within a general quantitative CLT framework, it proves a central limit theorem for the count $F_n(\eta)$ of extreme seeds in the growing window $W_n=[-n,n]^d$, with variance of order $n^d$ and convergence rates $d_W$ and $d_K$ of order $n^{-d/2}$ (Kolmogorov rate shown to be optimal). Tail bounds for the coverage time and variance lower bounds underpin the stabilization analysis and enable the explicit rate claims. The $\beta'$-model requires a stronger condition $\beta>5d+1$ due to non-integrable interaction terms, and the Gaussian model is treated as a scaling limit.Together, the results extend previous quantitative CLTs for the $\beta$-model to the $\beta'$ and Gaussian models and provide a robust methodology for further weighted-functionals of extreme points.

Abstract

We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the $β$-, $β'$- and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point processes in space-time $\mathbb{R}^d \times \mathbb{R}$, where some of the points of the process give rise to a cell in $\mathbb{R}^d$, known as extreme points, while the other points produce an empty cell. Using the notion of region-stabilization, we derive quantitative central limit theorems with presumably optimal rates of convergence for the number of extreme points of $β$-, $β'$- and Gaussian-Voronoi tessellations in a growing window $W_n=[-n,n]^d$ as $n\to\infty$. Our bounds improve and extend previously known results by Schreiber and Yukich (2008) for the $β$-model, and are the first quantitative results for the $β'$- and Gaussian models.

Gaussian approximation for Extreme Points in Laguerre tessellations

TL;DR

The article analyzes Gaussian approximations for the number of extreme points in Poisson-Laguerre tessellations across -, -, and Gaussian-Voronoi models. Using region stabilization within a general quantitative CLT framework, it proves a central limit theorem for the count of extreme seeds in the growing window , with variance of order and convergence rates and of order (Kolmogorov rate shown to be optimal). Tail bounds for the coverage time and variance lower bounds underpin the stabilization analysis and enable the explicit rate claims. The -model requires a stronger condition due to non-integrable interaction terms, and the Gaussian model is treated as a scaling limit.Together, the results extend previous quantitative CLTs for the -model to the and Gaussian models and provide a robust methodology for further weighted-functionals of extreme points.

Abstract

We consider Gaussian approximation in three particular models of Poisson-Laguerre tessellations, namely, the -, - and Gaussian-Voronoi tessellations. The tessellations are constructed based on inhomogeneous Poisson point processes in space-time , where some of the points of the process give rise to a cell in , known as extreme points, while the other points produce an empty cell. Using the notion of region-stabilization, we derive quantitative central limit theorems with presumably optimal rates of convergence for the number of extreme points of -, - and Gaussian-Voronoi tessellations in a growing window as . Our bounds improve and extend previously known results by Schreiber and Yukich (2008) for the -model, and are the first quantitative results for the - and Gaussian models.
Paper Structure (8 sections, 5 theorems, 178 equations, 2 figures)

This paper contains 8 sections, 5 theorems, 178 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. A quantitative CLT for stabilizing Poisson functionals
  5. Properties of Poisson-Laguerre tessellation
  6. Tail bounds for the coverage time
  7. Variance lower bounds
  8. Proof of Theorem \ref{['thm:1']}

Key Result

Theorem 1.1

Let $\eta$ be either (a) $\eta_\beta$ with $\beta>-1$, or (b) $\eta_\beta'$ with $\beta>5d+1$, or (c) $\widetilde{\eta}$. Then there exists $0 < C_1\le C_2<\infty$ depending only on $d,\beta,\gamma$ such that Moreover, for $N \sim \mathcal{N}(0,1)$, there exists a constant $C \in (0,\infty)$ depending only on $d,\beta,\gamma$ such that for all $n \in \mathbb{N}$, for $d \in \{d_W,d_K\}$. Moreov

Figures (2)

  • Figure 1: Left: $\beta$-Voronoi tessellation in $\mathbb{R}^2$ with $\beta=5$. Middle: $\beta'$-Voronoi tessellation in $\mathbb{R}^2$ with $\beta=2.5$. Right: Gaussian-Voronoi tessellation in $\mathbb{R}^2$. The corresponding centers of growth are marked in blue.
  • Figure 2: Illustration of the region $D((v_2,h_2), T(v_2,h_2),\mathcal{M})$ for some point configuration $\mathcal{M}$ containing these three points such that the height $T(v_2, h_2)$ is the coverage time of the cell of $(v_2,h_2)$ defined at \ref{['eq:covertime1']}

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2: Voronoi tessellation
  • Remark 1.3: Range of $\beta$ in Theorem \ref{['thm:1']} for the $\beta'$ model
  • Remark 1.4: Comparison with Johnson-Mehl tessellations
  • Remark 1.5: Extensions to weghted functionals of extreme points
  • Theorem 2.1: BM22, Theorem 2.1
  • Remark 2.2: Applicability in Laguerre tessellations
  • proof : Proof of Variance upper bound in Theorem \ref{['thm:KolBd']}
  • Lemma 3.1
  • proof
  • ...and 6 more