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June 2013 Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains
Lothar Heinrich, Malte Spiess
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Adv. in Appl. Probab. 45(2): 312-331 (June 2013). DOI: 10.1239/aap/1370870120

Abstract

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.

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Lothar Heinrich. Malte Spiess. "Central limit theorems for volume and surface content of stationary Poisson cylinder processes in expanding domains." Adv. in Appl. Probab. 45 (2) 312 - 331, June 2013. https://doi.org/10.1239/aap/1370870120

Information

Published: June 2013
First available in Project Euclid: 10 June 2013

zbMATH: 1282.60012
MathSciNet: MR3102453
Digital Object Identifier: 10.1239/aap/1370870120

Subjects:
Primary: 60D05, 60F05
Secondary: 60F10, 60G55

Rights: Copyright © 2013 Applied Probability Trust

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Vol.45 • No. 2 • June 2013
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