Central limit theorems for Poisson hyperplane tessellations

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 all earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640-656] for intersection points of rnotion-invariany Poisson line processes in R-2. Our proof is based oil 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. ill all increasing spherical region.