We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in ℝd. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640–656] for intersection points of motion-invariant Poisson line processes in ℝ2. Our proof is based on Hoeffding’s decomposition of U-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the “method of moments” used in [Adv. in Appl. Probab. 30 (1998) 640–656] to treat the case d=2. Moreover, we extend our central limit theorem in several directions. First we consider k-flat processes induced by Poisson hyperplane processes in ℝd for 0≤k≤d−1. Second we derive (asymptotic) confidence intervals for the intensities of these k-flat processes and, third, we prove multivariate central limit theorems for the d-dimensional joint vectors of numbers of k-flats and their k-volumes, respectively, in an increasing spherical region.
Publié le : 2006-05-14
Classification:
Poisson hyperplane process,
point process,
k-flat intersection process,
U-statistic,
Hoeffding’s decomposition,
central limit theorem,
confidence interval,
long-range dependence,
60D05,
60F05,
62F12
@article{1151592255,
author = {Heinrich, Lothar and Schmidt, Hendrik and Schmidt, Volker},
title = {Central limit theorems for Poisson hyperplane tessellations},
journal = {Ann. Appl. Probab.},
volume = {16},
number = {1},
year = {2006},
pages = { 919-950},
language = {en},
url = {http://dml.mathdoc.fr/item/1151592255}
}
Heinrich, Lothar; Schmidt, Hendrik; Schmidt, Volker. Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Probab., Tome 16 (2006) no. 1, pp. 919-950. http://gdmltest.u-ga.fr/item/1151592255/