Limit theorems for geometric functionals of Gibbs point processes

T. Schreiber; J. E. Yukich

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1158-1182
  • ISSN: 0246-0203

Abstract

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Observations are made on a point process 𝛯 in ℝ d in a window Q λ of volume λ . The observation, or ‘score’ at a point x , here denoted ξ ( x , 𝛯 ) , is a function of the points within a random distance of x . When the input 𝛯 is a Poisson or binomial point process, the large λ limit theory for the total score ∑ x ∈ 𝛯 ∩ Q λ ξ ( x , 𝛯 ∩ Q λ ) , when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input 𝛯 . The proofs use perfect simulation of Gibbs point processes to establish their mixing properties. The general limit results are applied to random sequential packing and spatial birth growth models, Voronoi and other Euclidean graphs, percolation models, and quantization problems involving Gibbsian input.

How to cite

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Schreiber, T., and Yukich, J. E.. "Limit theorems for geometric functionals of Gibbs point processes." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1158-1182. <http://eudml.org/doc/271984>.

@article{Schreiber2013,
abstract = {Observations are made on a point process $\varXi $ in $\mathbb \{R\}^\{d\}$ in a window $Q_\{\lambda \}$ of volume $\{\lambda \}$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,\varXi )$, is a function of the points within a random distance of $x$. When the input $\varXi $ is a Poisson or binomial point process, the large $\{\lambda \}$ limit theory for the total score $\sum _\{x\in \varXi \cap Q_\{\lambda \}\}\xi (x,\varXi \cap Q_\{\lambda \})$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $\varXi $. The proofs use perfect simulation of Gibbs point processes to establish their mixing properties. The general limit results are applied to random sequential packing and spatial birth growth models, Voronoi and other Euclidean graphs, percolation models, and quantization problems involving Gibbsian input.},
author = {Schreiber, T., Yukich, J. E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Perfect simulation; Gibbs point processes; exponential mixing; gaussian limits; hard core model; random packing; geometric graphs; Gibbs–Voronoi tessellations; quantization; locally finite set; geometric functionals; central limit theorem; laws of large mumbers},
language = {eng},
number = {4},
pages = {1158-1182},
publisher = {Gauthier-Villars},
title = {Limit theorems for geometric functionals of Gibbs point processes},
url = {http://eudml.org/doc/271984},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Schreiber, T.
AU - Yukich, J. E.
TI - Limit theorems for geometric functionals of Gibbs point processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1158
EP - 1182
AB - Observations are made on a point process $\varXi $ in $\mathbb {R}^{d}$ in a window $Q_{\lambda }$ of volume ${\lambda }$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,\varXi )$, is a function of the points within a random distance of $x$. When the input $\varXi $ is a Poisson or binomial point process, the large ${\lambda }$ limit theory for the total score $\sum _{x\in \varXi \cap Q_{\lambda }}\xi (x,\varXi \cap Q_{\lambda })$, when properly scaled and centered, is well understood. In this paper we establish general laws of large numbers, variance asymptotics, and central limit theorems for the total score for Gibbsian input $\varXi $. The proofs use perfect simulation of Gibbs point processes to establish their mixing properties. The general limit results are applied to random sequential packing and spatial birth growth models, Voronoi and other Euclidean graphs, percolation models, and quantization problems involving Gibbsian input.
LA - eng
KW - Perfect simulation; Gibbs point processes; exponential mixing; gaussian limits; hard core model; random packing; geometric graphs; Gibbs–Voronoi tessellations; quantization; locally finite set; geometric functionals; central limit theorem; laws of large mumbers
UR - http://eudml.org/doc/271984
ER -

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