Limit theorems for geometric functionals of Gibbs point processes
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, p. 1158-1182

Observations are made on a point process 𝛯 in 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 x𝛯Q λ ξ(x,𝛯Q 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.

On observe un processus ponctuel 𝛯 dans d dans une fenêtre Q λ de volume λ. L’observation en un point x que l’on note ξ(x,𝛯) est une fonction des points situés à une distance aléatoire de x. Quand 𝛯 est un processus de Poisson ponctuel ou Binomial, la limite pour λ grand de la somme totale x𝛯Q λ ξ(x,𝛯Q λ ) (convenablement recentrée et normalisée) est bien comprise. Dans ce papier, nous étudions cette somme totale quand 𝛯 est Gibbsien et prouvons la loi des grands nombres, la variance asymptotique et un théorème de la limite centrale. Les preuves reposent sur la simulation parfaite de processus ponctuels Gibbsiens pour établir leurs propriétés de mélange. Ces résultats généraux sont appliqués dans différents contextes comme des modèles de croissance et de percolation, des graphes de Voronoi et des problèmes de quantification pour des entrées Gibbsiennes.

DOI : https://doi.org/10.1214/12-AIHP500
Classification:  60F05,  60G55,  60D05
Keywords: Perfect simulation, Gibbs point processes, exponential mixing, gaussian limits, hard core model, random packing, geometric graphs, Gibbs-Voronoi tessellations, quantization
@article{AIHPB_2013__49_4_1158_0,
     author = {Schreiber, T. and Yukich, Joseph E.},
     title = {Limit theorems for geometric functionals of Gibbs point processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {4},
     year = {2013},
     pages = {1158-1182},
     doi = {10.1214/12-AIHP500},
     mrnumber = {3127918},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_4_1158_0}
}
Limit theorems for geometric functionals of Gibbs point processes. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 4, pp. 1158-1182. doi : 10.1214/12-AIHP500. http://www.numdam.org/item/AIHPB_2013__49_4_1158_0/

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