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 $\mathbb{R}^d$. 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 $\mathbb{R}^2$. 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 $\mathbb{R}^d$ for $0\le k\le d-1$. 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.Comment: Published at http://dx.doi.org/10.1214/105051606000000033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

“…, d} of a stationary Poisson k-flat process in R d . We thereby considerably extend the results available in the literature (see [4] and [5]) for the number of intersections and the intersection volume. In our theory we can allow for very general geometric functionals; for example, we do not require them to be additive, translation-invariant, or homogeneous.…”

Second‐order properties of the point process of nodes in a stationary Voronoi tessellation

“…Making use of the β-mixing property of a stationary PVT expressed in terms of a suitable β-mixing coefficient having exponential decay (see Theorem 2.1 in Heinrich (1994)), we may deduce from Corollary 2.3 in Heinrich (1994) that the unbiased estimator λ V,n = Ψ V (A n )/|A n | for λ V is asymptotically normally distributed as n → ∞ with mean λ V = κ d λ and the asymptotic variance σ 2 V = λκ d 1 + κ d σ 2 d , see (2.5) and (5.8). A corresponding result for the number of nodes induced by a d-dimensional Poisson hyperplane tessellation (which is not β-mixing) has been recently proved in Heinrich et al (2006). For another approach to central limit theorems for random tessellations the reader is referred to Penrose and Yukich (2001).…”

“…In the special cases d = 2 and d = 3 formula (3.15) is seen from the shape of the corresponding K-function which is obtained by applying Slivnyak's theorem to the Poisson line and plane process, seeArns et al(2005). InHeinrich et al (2006) a closed-form expression for the varianceVarΨ H (b(o, r)) as polynomial of degree 2d − 1 in r and the relationship…”

Some inequalities for chord power integrals of parallelotopes

“…Let us sketch a typical example -another one is discussed in [6]. To be precise we need some further notation, for details the reader is referred to [4].…”

“…is asymptotically normally distributed with variance σ 2 k,d (λ, K), see [4]. The dependence of σ 2 k,d (λ, K) on the shape of…”