Clt S 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 an earlier one proved by Paroux [Adv. for intersection points of motion-invariant Poisson line processes in R 2. Our proof is based on Hoeffd-ing'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. 1. Introduction. Central limit theorems (briefly CLTs) for models of stochastic geometry have been considered in various papers. For example, [1] and [24] investigate CLTs for Poisson–Voronoi and Poisson line tessellations in the Euclidean plane, respectively. More general CLTs for Poisson–Voronoi tessellations in the d-dimensional Euclidean space R d have been established in [12] and [25]. In [9], normal approximations are given for some mean-value estimates of absolutely regular (β-mixing) tessellations. A CLT for stationary tessellations with random inner cell structures has been derived in [13]. Furthermore, CLTs and related asymptotic properties for the empirical volume fraction of stationary random sets in R d are examined in [2, 5, 16].