Gaussian limits for random measures in geometric probability

Abstract

We establish Gaussian limits for general measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general results are used to deduce central limit theorems for measures induced by random graphs (nearest neighbor, Voronoi and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth-growth models) and statistics of germ-grain models.Comment: Published at http://dx.doi.org/10.1214/105051604000000594 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Process level moderate deviations for stabilizing functionals

“…One crucial assumption imposed on ξ throughout this paper is the so-called exponential stabilization, see [1,[12][13][14]. We say that ξ is stabilizing (at intensity τ ) if for each x ∈ R d there exists an a.s. finite random variable R(x) := R ξ (x, P) (a radius of stabilization) and ξ ∞ (x) := ξ ∞ (x, P) (the limit of ξ) such that, with probability one, ξ(x, (P ∩ B R(x) (x)) ∪ σ) = ξ ∞ (x) for all locally finite σ ⊆ R d \ B R(x) (x).…”

“…In [1,[12][13][14] a lot of examples of stabilizing functionals are discussed. In Section 2 we will focus on random sequential packing models, birth-growth models, germ-grain models and nearest neighbor graphs.…”

“…From [1,16] we know that ξ satisfies the exponential stabilization condition (E). Moreover, by Section 2 and Sections 6.1 and 6.2 of [2] we see that ξ satisfies both the (L) and (L') conditions.…”

Nearest-neighbor graphs on the cantor set

“…Recall thatL is log 2 periodic by Step 2. To show thatL is continuous for all α > 0, it is enough to show, by periodicity, that L is continuous on [1,2]. Since …”

The Transparent Dead Leaves Model

“…Similar random functionals interpreted as the sum of contributions from each point of a (possibly marked) point process {x i } ⊂ R d appear in several contexts in stochastic geometry. In particular, general central limit theorems hold when the intensity of the point process {x i } tends to infinity, see e.g [3,21]. Note however that our framework is different since the Poisson process has an additional time component t i , and consequently, there is always an infinite number of leaves (t i , x i , X i , a i ) influencing the value f (y) (as will be clarified by Proposition 3).…”