The Annals of Applied Probability

Surface order scaling in stochastic geometry

J. E. Yukich

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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.

Article information

Ann. Appl. Probab., Volume 25, Number 1 (2015), 177-210.

First available in Project Euclid: 16 December 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Poisson–Voronoi tessellation Poisson–Voronoi volume estimator Poisson–Voronoi surface area estimator maximal points


Yukich, J. E. Surface order scaling in stochastic geometry. Ann. Appl. Probab. 25 (2015), no. 1, 177--210. doi:10.1214/13-AAP992.

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