Limit theorems for geometric functionals of Gibbs point processes

Schreiber, T.; Yukich, J. E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1158-1182.

On observe un processus ponctuel $𝛯$ dans ${ℝ}^d$ dans une fenêtre $Q_{\lambda }$ de volume $\lambda$. L’observation en un point $x$ que l’on note $\xi (x,𝛯)$ 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 $\lambda$ grand de la somme totale ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$ (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.

Observations are made on a point process $𝛯$ in ${ℝ}^d$ in a window $Q_{\lambda }$ of volume $\lambda$. The observation, or ‘score’ at a point $x$, here denoted $\xi (x,𝛯)$, 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 $\lambda$ limit theory for the total score ${\sum }_{x\in 𝛯\cap Q_{\lambda }}\xi (x,𝛯\cap Q_{\lambda })$, 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 $𝛯$. 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.

[1] A. J. Baddeley and M. N. M. Van Lieshout. Area interaction point processes. Ann. Inst. Statist. Math. 47 (1995) 601-619. | MR

[2] Y. Baryshnikov and J. E. Yukich. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005) 213-253. | MR

[3] J. A. Bucklew and G. Wise. Multidimensional asymptotic quantization theory with $r$th power distortion measures. IEEE Trans. Inf. Th. 28 (1982) 239-247. | MR

[4] P. Cohort. Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004) 118-143. | MR

[5] D. J. Daley and D. Vere-Jones. An Introduction to the Theory of Point Processes, vol. II, 2nd edition. Springer, New York, 2008. | MR

[6] D. Dereudre and F. Lavancier. Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction. Comput. Statist. Data Anal. 55 (2011) 498-519. | MR

[7] A. Dvoretzky and H. Robbins. 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

[8] J. W. Evans. Random and cooperative adsorption. Rev. Modern Physics 65 (1993) 1281-1329.

[9] R. Fernández, P. Ferrari and N. Garcia. Measures on contour, polymer or animal models. A probabilistic approach. Markov Process. Related Fields 4 (1998) 479-497. | MR

[10] R. Fernández, P. Ferrari and N. Garcia. Loss network representation of Peierls contours. Ann. Probab. 29 (2001) 902-937. | MR

[11] R. Fernández, P. Ferrari and N. Garcia. Perfect simulation for interacting point processes, loss networks and Ising models. Stochastic Process. Appl. 102 (2002) 63-88. | MR

[12] S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. Lecture Notes in Mathemmatics 1730. Springer, Berlin, 2000. | MR

[13] G. Grimmett. Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften 321. Springer, Berlin, 1999. | MR

[14] O. Häggström and R. Meester. Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 (1996) 295-315. | MR

[15] P. Hall. On continuum percolation. Ann. Probab. 13 (1985) 1250-1266. | MR

[16] I. Kozakova, R. Meester and S. Nanda. The size of components in continuum nearest-neighbor graphs. Ann. Probab. 34 (2006) 528-538. | MR

[17] C. Lautensack and T. Sych. 3D image analysis of open foams using random tessellations. Image Anal. Stereo. 25 (2006) 87-93.

[18] I. Molchanov and N. Tontchev. Optimal Poisson quantisation. Stat. Probab. Letters 77 (2007) 1123-1132. | MR

[19] J. Møller and R. Waagepetersen. Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall, Boca Raton, 2004. | MR

[20] M. D. Penrose. Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007) 989-1035. | MR

[21] M. D. Penrose. Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13 (2007) 1124-1150. | MR

[22] M. D. Penrose and J. E. Yukich. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001) 1005-1041. | MR

[23] M. D. Penrose and J. E. Yukich. Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 (2002) 272-301. | MR

[24] M. D. Penrose and J. E. Yukich. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003) 277-303. | MR

[25] M. D. Penrose and J. E. Yukich. 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

[26] A. Rényi. 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] D. Ruelle. Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 (1970) 127-159. | MR

[28] T. Schreiber, M. D. Penrose and J. E. Yukich. Gaussian limits for multidimensional random sequential packing at saturation. Comm. Math. Phys. 272 (2007) 167-183. | MR

[29] D. Stoyan, W. Kendall and J. Mecke. Stochastic Geometry and Its Applications, 2nd edition. Wiley, New York, 1995. | MR

[30] J. E. Yukich. Limit theorems for multi-dimensional random quantizers. Electron. Commun. Probab. 13 (2008) 507-517. | MR

[31] J. E. Yukich. 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] P. L. Zador. Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 (1982) 139-149. | MR