Abstract
For the Poisson line process the empirical polygon is defined thanks to ergodicity and laws of large numbers for some characteristics, such as the number of edges, the perimeter, the area, etc... One also has, still thanks to the ergodicity of the Poisson line process, a canonical relation between this empirical polygon and the polygon containing a given point. The aim of this paper is to provide numerical simulations for both methods: in a previous paper (Paroux, Advances in Applied Probability, 30:640–656, 1998) one of the authors proved central limit theorems for some geometrical quantities associated with this empirical Poisson polygon, we provide numerical simulations for this phenomenon which generates billions of polygons at a small computational cost. We also give another direct simulation of the polygon containing the origin, which enables us to give further values for empirical moments of some geometrical quantities than the ones that were known or computed in the litterature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Calka, “De nouveaux résultats sur la géométrie des mosaïques de Poisson-Voronoi et des mosaïques poissoniennes d’hyperplans. Et́ude du modèle de fissuration de Rényi-Widon,” Thèse de doctorat, Université Lyon 1, 2002.
R. Cowan, “The use of the ergodic theorems in random geometry,” Supplement to Advances in Applied Probability vol. 10, pp. 47–57, 1978.
R. Cowan, “Properties of ergodic random mosaic processes,” Mathematische Nachrichten vol. 97, pp. 89–102, 1980.
I. Crain and R. E. Miles, “Monte Carlo estimates of the distributions of the random polygons determined by random lines in the plan,” Journal of Statistical Computation and Simulation vol. 4, pp. 293–325, 1976.
L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag: New York, Berlin, Heidelberg, 1986.
E. I. George, “Sampling random polygons,” Journal of Applied Probability vol. 24, pp. 557–573, 1987.
A. Goldman, “Le spectre de certaines mosaïques poissoniennes du plan et l’enveloppe convexe du pont brownien,” Probability Theory and Related Fields vol. 105, pp. 57–83, 1996.
J. F. C. Kingman, Poisson Processes, Oxford University Press: New York, 1993.
R. E. Miles, “Random polygons determined by random lines in a plane I,” Proceedings of the National Academy of Sciences of the United States of America vol. 52, pp. 901–907, 1964a.
R. E. Miles, “Random polygons determined by random lines in a plane II,” Proceedings of the National Academy of Sciences of the United States of America vol. 52, pp. 1157–1160, 1964b.
K. Paroux, “Théorèmes centraux limites pour les processus poissoniens de droites dans le plan et questions de convergence pour le modèle booléen de l’espace euclidien,” Thèse de doctorat, Université Lyon 1, 1997.
K. Paroux, “Quelques théorèmes centraux limites pour les processus poissoniens de droites du plan,” Advances in Applied Probability vol. 30, pp. 640–656, 1998.
H. Solomon, Geometric Probability, SIAM: Philadelphia, 1978.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the PSMN at ENS-Lyon.
Rights and permissions
About this article
Cite this article
Michel, J., Paroux, K. Empirical Polygon Simulation and Central Limit Theorems for the Homogenous Poisson Line Process. Methodol Comput Appl Probab 9, 541–556 (2007). https://doi.org/10.1007/s11009-006-9009-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-006-9009-z