Normal Approximation for Coverage Models over Binomial Point Processes by Larry Goldstein

We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of n points in a toroidal spatial region of volume n. The proof is based on Stein's method via size-biased couplings. 1. Introduction. Given a collection of n independent uniformly distributed random points in a d-dimensional cube of volume n (the so-called binomial point process), let V denote the (random) total volume of the union of interpenetrating balls of fixed radius ρ centered at these points, and let S denote the number of balls of radius ρ/2 (centered at the same set of points) which are singletons, that is, do not overlap any other such ball. These variables are fundamental topics of interest in the stochastic geometry of coverage processes and random geometric graphs [9, 10, 13, 18]. As n → ∞ with ρ fixed (the so-called thermodynamic limit), both V and S are known to satisfy a central limit theorem (CLT) [12, 13, 16]. In the present work we provide associated Berry–Esseen type results; that is, we show under periodic boundary conditions that the cumulative distribution functions converge to that of the normal at the same O(n −1/2) rate as for a sum of n independent identically distributed variables, and provide bounds on the quality of the normal approximation for finite n. Were we to consider instead a Poisson-distributed number of points, that is, a Poisson point process instead of a binomial one, both of our variables of interest could be expressed as sums of locally dependent random variables, and thereby Berry–Esseen type bounds could be (and have been) obtained by known methods [1, 8, 15, 17]. But with a nonrandom number of points, the local dependence is lost and the de-Poissonization arguments in [13, 16] do not provide error bounds for the de-Poissonized CLTs. The early work of Moran [11, 12] on V was in response to queries in the statistical physics literature (including the well-known