Gaussian approximation for Extreme Points in Laguerre tessellations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Bhattacharjee, Chinmoy Gusakova, Anna |
| author_facet | Bhattacharjee, Chinmoy Gusakova, Anna |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaussian approximation for Extreme Points in Laguerre tessellations Bhattacharjee, Chinmoy Gusakova, Anna Probability 60D05, 60F05 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. |
| title | Gaussian approximation for Extreme Points in Laguerre tessellations |
| topic | Probability 60D05, 60F05 |
| url | https://arxiv.org/abs/2510.21665 |