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Central Limit Theorems for Volume and Surface Content of Stationary Poisson Cylinder Processes in Expanding Domains

Part of: Limit theorems Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Lothar Heinrich  and
Malte Spiess
Lothar Heinrich*
Affiliation:
Augsburg University
Malte Spiess*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Mathematics, Augsburg University, D-86135 Augsburg, Germany. Email address: [email protected]
∗∗ Postal address: Institute of Stochastics, Ulm University, Helmholtzstr. 18, D-89069 Ulm, Germany. Email address: [email protected]
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Abstract

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A stationary Poisson cylinder process in the d-dimensional Euclidean space is composed of a stationary Poisson process of k-flats (0 ≤ k ≤ d−1) which are dilated by independent and identically distributed random compact cylinder bases taken from the corresponding (d−k)-dimensional orthogonal complement. If the second moment of the (d−k)-volume of the typical cylinder base exists, we prove asymptotic normality of the d-volume of the union set of Poisson cylinders that covers an expanding star-shaped domain ϱ W as ϱ grows unboundedly. Due to the long-range dependencies within the union set of cylinders, the variance of its d-volume in ϱ W increases asymptotically proportional to the (d+k) th power of ϱ. To obtain the exact asymptotic behaviour of this variance, we need a distinction between discrete and continuous directional distributions of the typical k-flat. A corresponding central limit theorem for the surface content is stated at the end.

Keywords

Independently marked Poisson processtruncated typical cylinderdirection spacevolume fractionmoment convergence theoremasymptotic variancelong-range dependencehigher-order (mixed) cumulant

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 2 , June 2013 , pp. 312 - 331
Copyright
© Applied Probability Trust 

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