Observations are made on a point process in in a window of volume . The observation, or ‘score’ at a point , here denoted , is a function of the points within a random distance of . When the input is a Poisson or binomial point process, the large limit theory for the total score , 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 . 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 dans dans une fenêtre de volume . L’observation en un point que l’on note est une fonction des points situés à une distance aléatoire de . Quand est un processus de Poisson ponctuel ou Binomial, la limite pour grand de la somme totale (convenablement recentrée et normalisée) est bien comprise. Dans ce papier, nous étudions cette somme totale quand 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.
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, J. E.}, title = {Limit theorems for geometric functionals of {Gibbs} point processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1158--1182}, publisher = {Gauthier-Villars}, volume = {49}, number = {4}, year = {2013}, doi = {10.1214/12-AIHP500}, mrnumber = {3127918}, language = {en}, url = {http://www.numdam.org/articles/10.1214/12-AIHP500/} }
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 SP - 1158 EP - 1182 VL - 49 IS - 4 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/12-AIHP500/ DO - 10.1214/12-AIHP500 LA - en ID - AIHPB_2013__49_4_1158_0 ER -
%0 Journal Article %A Schreiber, T. %A Yukich, J. E. %T Limit theorems for geometric functionals of Gibbs point processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 1158-1182 %V 49 %N 4 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/12-AIHP500/ %R 10.1214/12-AIHP500 %G en %F AIHPB_2013__49_4_1158_0
Schreiber, T.; Yukich, J. E. 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/articles/10.1214/12-AIHP500/
[1] Area interaction point processes. Ann. Inst. Statist. Math. 47 (1995) 601-619. | MR
and .[2] Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213-253. | MR
and .[3] Multidimensional asymptotic quantization theory with th power distortion measures. IEEE Trans. Inf. Th. 28 (1982) 239-247. | MR
and .[4] Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004) 118-143. | MR
.[5] An Introduction to the Theory of Point Processes, vol. II, 2nd edition. Springer, New York, 2008. | MR
and .[6] Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction. Comput. Statist. Data Anal. 55 (2011) 498-519. | MR
and .[7] On the “parking” problem. MTA Mat Kut. Int. Kz̈l. (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 9 (1964) 209-225. | MR
and .[8] Random and cooperative adsorption. Rev. Modern Physics 65 (1993) 1281-1329.
.[9] Measures on contour, polymer or animal models. A probabilistic approach. Markov Process. Related Fields 4 (1998) 479-497. | MR
, and .[10] Loss network representation of Peierls contours. Ann. Probab. 29 (2001) 902-937. | MR
, and .[11] Perfect simulation for interacting point processes, loss networks and Ising models. Stochastic Process. Appl. 102 (2002) 63-88. | MR
, and .[12] Foundations of Quantization for Probability Distributions. Lecture Notes in Mathemmatics 1730. Springer, Berlin, 2000. | MR
and .[13] Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin, 1999. | MR
.[14] Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 (1996) 295-315. | MR
and .[15] On continuum percolation. Ann. Probab. 13 (1985) 1250-1266. | MR
.[16] The size of components in continuum nearest-neighbor graphs. Ann. Probab. 34 (2006) 528-538. | MR
, and .[17] 3D image analysis of open foams using random tessellations. Image Anal. Stereo. 25 (2006) 87-93.
and .[18] Optimal Poisson quantisation. Stat. Probab. Letters 77 (2007) 1123-1132. | MR
and .[19] Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall, Boca Raton, 2004. | MR
and .[20] Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007) 989-1035. | MR
.[21] Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007) 1124-1150. | MR
.[22] Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005-1041. | MR
and .[23] Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272-301. | MR
and .[24] Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277-303. | MR
and .[25] Normal approximation in geometric probability. In Stein's Method and Applications 37-58. A. D. Barbour and Louis H. Y. Chen (Eds.). Lecture Note Series. Institute for Mathematical Sciences, National University of Singapore 5. Singapore Univ. Press, Singapore, 2005. Available at http://www.lehigh.edu/~jey0/publications.html. | MR
and .[26] On a one-dimensional random space-filling problem. MTA Mat Kut. Int. Kz̈l. (Publications of the Math. Res. Inst. of the Hungarian Academy of Sciences) 3 (1958) 109-127. | MR
.[27] Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 (1970) 127-159. | MR
.[28] Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys. 272 (2007) 167-183. | MR
, and .[29] Stochastic Geometry and Its Applications, 2nd edition. Wiley, New York, 1995. | MR
, and .[30] Limit theorems for multi-dimensional random quantizers. Electron. Commun. Probab. 13 (2008) 507-517. | MR
.[31] Limit theorems in discrete stochastic geometry. In Stochastic Geometry, Spatial Statistics and Random Fields 239-275. E. Spodarev (Ed.). Lecture Notes in Mathematics 2068. Springer, Berlin, 2013. | MR
.[32] Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 (1982) 139-149. | MR
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