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February 2015 Surface order scaling in stochastic geometry
J. E. Yukich
Ann. Appl. Probab. 25(1): 177-210 (February 2015). DOI: 10.1214/13-AAP992

Abstract

Let ${{ \mathcal{P}}}_{{\lambda}}:={{ \mathcal{P}}}_{{\lambda}{\kappa}}$ denote a Poisson point process of intensity ${\lambda}{\kappa}$ on $[0,1]^{d}$, $d\geq2$, with ${\kappa}$ a bounded density on $[0,1]^{d}$ and ${\lambda}\in(0,\infty)$. Given a closed subset ${ \mathcal{M}}\subset[0,1]^{d}$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi(x,{{ \mathcal{P}}}_{{\lambda}},{ \mathcal{M}})$, where the score function $\xi$ vanishes unless the input $x$ is close to ${ \mathcal{M}}$ and where $\xi$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in{{ \mathcal{P}}}_{{\lambda}}}\xi({\lambda}^{1/d}x,{\lambda}^{1/d}{{ \mathcal{P}}}_{{\lambda}},{\lambda}^{1/d}{ \mathcal{M}})$ as ${\lambda}\to\infty$. When ${ \mathcal{M}}$ is of class $C^{2}$, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order $\mathrm{Vol} ({\lambda}^{1/d}{ \mathcal{M}})$. We use the general results to deduce variance asymptotics and central limit theorems for statistics arising in stochastic geometry, including Poisson–Voronoi volume and surface area estimators, answering questions in Heveling and Reitzner [Ann. Appl. Probab. 19 (2009) 719–736] and Reitzner, Spodarev and Zaporozhets [Adv. in Appl. Probab. 44 (2012) 938–953]. The general results also yield the limit theory for the number of maximal points in a sample.

Citation

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J. E. Yukich. "Surface order scaling in stochastic geometry." Ann. Appl. Probab. 25 (1) 177 - 210, February 2015. https://doi.org/10.1214/13-AAP992

Information

Published: February 2015
First available in Project Euclid: 16 December 2014

zbMATH: 1356.60041
MathSciNet: MR3297770
Digital Object Identifier: 10.1214/13-AAP992

Subjects:
Primary: 60F05
Secondary: 60D05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 1 • February 2015
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