Surface order scaling in stochastic geometry

Abstract

Let ${\mathcal{P}}_{\lambda }:={\mathcal{P}}_{\lambda \kappa }$ denote a Poisson point process of intensity $\lambda \kappa$ on $[0,1{]}^d,d\ge 2$, 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.

Published in at http://dx.doi.org/10.1214/13-AAP992 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Gaussian limits for random measures in geometric probability
• Laws of large numbers in stochastic geometry with statistical applications
• Nonparametric estimation of surface integrals
• Poisson--Voronoi approximation
• Asymptotic theory for statistics of the Poisson--Voronoi approximation

Cited by in corpus (3)

• Limit theory of isolated and extreme points in hyperbolic random geometric graphs
• Asymptotic theory for statistics of the Poisson--Voronoi approximation
• New Kolmogorov bounds for functionals of binomial point processes