Central limit theorems for Poisson hyperplane tessellations

Abstract

We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in ℝ^d. 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 ℝ^d 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.