Surface order scaling in stochastic geometry

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.