Annals of Applied Probability

Central limit theorems for Poisson hyperplane tessellations

Lothar Heinrich, Hendrik Schmidt, and Volker Schmidt

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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≤kd−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.

Article information

Ann. Appl. Probab., Volume 16, Number 2 (2006), 919-950.

First available in Project Euclid: 29 June 2006

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60F05: Central limit and other weak theorems 62F12: Asymptotic properties of estimators

Poisson hyperplane process point process k-flat intersection process U-statistic Hoeffding’s decomposition central limit theorem confidence interval long-range dependence


Heinrich, Lothar; Schmidt, Hendrik; Schmidt, Volker. Central limit theorems for Poisson hyperplane tessellations. Ann. Appl. Probab. 16 (2006), no. 2, 919--950. doi:10.1214/105051606000000033.

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