## The Annals of Applied Probability

### Surface order scaling in stochastic geometry

J. E. Yukich

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 16 December 2014

https://projecteuclid.org/euclid.aoap/1418740183

Digital Object Identifier
doi:10.1214/13-AAP992

Mathematical Reviews number (MathSciNet)
MR3297770

Zentralblatt MATH identifier
1356.60041

#### Citation

Yukich, J. E. Surface order scaling in stochastic geometry. Ann. Appl. Probab. 25 (2015), no. 1, 177--210. doi:10.1214/13-AAP992. https://projecteuclid.org/euclid.aoap/1418740183

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