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Surface order scaling in stochastic geometry

Let $\mathcal{P}_λ:=\mathcal{P}_{λκ}$ denote a Poisson point process of intensity $λκ$ on $[0,1]^d,d\geq2$, with $κ$ a bounded density on $[0,1]^d$ and $λ\in(0,\infty)$. Given a closed subset $\mathcal{M}\subset[0,1]^d$ of Hausdorff dimension $(d-1)$, we consider general statistics $\sum_{x\in\mathcal{P}_λ}ξ(x,\mathcal{P} _λ,\mathcal{M})$, where the score function $ξ$ vanishes unless the input $x$ is close to $\mathcal{M}$ and where $ξ$ satisfies a weak spatial dependency condition. We give a rate of normal convergence for the rescaled statistics $\sum_{x\in\mathcal{ P}_λ}ξ(λ^{1/d}x,λ^{1/d}\mathcal{P}_λ,λ^{1/d}\mathcal{M})$ as $λ\to\infty$. 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 $\mathrm{Vol}(λ^{1/d}\mathcal{M})$. 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.