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Yukich, J. E.

Surface order scaling in stochastic geometry. (English) Zbl 1356.60041

Ann. Appl. Probab. 25, No. 1, 177-210 (2015).
Summary: Let \(\mathcal{P}_{\lambda}:=\mathcal{P}_{\lambda\kappa}\) denote a Poisson point process of intensity \(\lambda\kappa\) on \([0,1]^d\), \(d\geq 2\), 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 \(\text{Vol}(\lambda^{1/d}\mathcal{M})\). 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 [M. Heveling and M. Reitzner, Ann. Appl. Probab. 19, No. 2, 719–736 (2009; Zbl 1172.60003)] and [M. Reitzner et al., Adv. Appl. Probab. 44, No. 4, 938–953 (2012; Zbl 1280.60013)]. The general results also yield the limit theory for the number of maximal points in a sample.

Cited in 14 Documents

MSC:

60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

stochastic geometry; normal convergence; weak laws of large numbers; variance asymptotics; central limit theorems; Poisson-Voronoi tessellation; Poisson-Voronoi volume estimator; Poisson-Voronoi surface area estimator; maximal points

Citations:

Zbl 1172.60003; Zbl 1280.60013
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Full Text: DOI arXiv Euclid

References:

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