Limit theory for unbiased and consistent estimators of statistics of random tessellations

Flimmel, Daniela; Pawlas, Zbyněk; Yukich, J. E.

doi:10.1017/jpr.2020.4

Cited by 5 publications

(2 citation statements)

References 14 publications

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“…Proof. The stabilising part of the proof is inspired by [38], and the idea can also be used to show the stabilisation property for Laguerre tessellations, which are a generalisation of Voronoi tessellations, as in [23].…”

Section: Theorem 32 If Is a Homogeneous Poisson Point Process Thenmentioning

confidence: 99%

Normal approximation in total variation for statistics in geometric probability

Cong

Xia

2023

Adv. Appl. Probab.

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We use Stein’s method to establish the rates of normal approximation in terms of the total variation distance for a large class of sums of score functions of samples arising from random events driven by a marked Poisson point process on $\mathbb{R}^d$ . As in the study under the weaker Kolmogorov distance, the score functions are assumed to satisfy stabilisation and moment conditions. At the cost of an additional non-singularity condition, we show that the rates are in line with those under the Kolmogorov distance. We demonstrate the use of the theorems in four applications: Voronoi tessellations, k-nearest-neighbours graphs, timber volume, and maximal layers.

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Section: Theorem 32 If Is a Homogeneous Poisson Point Process Thenmentioning

confidence: 99%

Normal approximation in total variation for statistics in geometric probability

Cong

Xia

2023

Adv. Appl. Probab.

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“…More interestingly, one can try to prove these stabilization and asymptotic essential connectedness properties (which are fundamental for our approach) for the generalized Poisson–Voronoi weighted tessellation [21], including, as special cases, Laguerre and Johnson–Mehl tessellations. Note that in this latter paper, a different stabilization property is used to prove expectation and variance asymptotics, as well as central limit theorems for unbiased and asymptotically consistent estimators of geometric statistics of the typical cell.…”

Section: Model Extensions and Concluding Remarksmentioning

confidence: 99%

Continuum line-of-sight percolation on Poisson–Voronoi tessellations

et al. 2021

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In this work, we study a new model for continuum line-of-sight percolation in a random environment driven by the Poisson–Voronoi tessellation in the d-dimensional Euclidean space. The edges (one-dimensional facets, or simply 1-facets) of this tessellation are the support of a Cox point process, while the vertices (zero-dimensional facets or simply 0-facets) are the support of a Bernoulli point process. Taking the superposition Z of these two processes, two points of Z are linked by an edge if and only if they are sufficiently close and located on the same edge (1-facet) of the supporting tessellation. We study the percolation of the random graph arising from this construction and prove that a 0–1 law, a subcritical phase, and a supercritical phase exist under general assumptions. Our proofs are based on a coarse-graining argument with some notion of stabilization and asymptotic essential connectedness to investigate continuum percolation for Cox point processes. We also give numerical estimates of the critical parameters of the model in the planar case, where our model is intended to represent telecommunications networks in a random environment with obstructive conditions for signal propagation.

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