Surface Order Scaling in Stochastic Geometry

Let Pλ := Pλκ denote a Poisson point process of intensity λκ on [0,1]d , d ≥ 2, with κ a bounded density on [0,1]d and λ ∈ (0,∞). Given a closed subset M⊂ [0,1]d of Hausdorff dimension (d−1), we consider general statistics ∑ x∈Pλ ξ(x,Pλ,M), where the score function ξ vanishes unless the input x is close to M and where ξ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics ∑ x∈Pλ ξ(λx,λPλ,λM) as λ→∞. When M is of class C2, we obtain weak laws of large numbers and variance asymptotics for these statistics, showing that growth is surface order, that is, of order Vol(λ1/dM). 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.