Abstract
We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich (Ann. Appl. Probab. 11(4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.
Highlights
The paper introduces a new approach to establish central limit theorems (CLT) for functionals on binomial point processes and Poisson point processes
We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich
Functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions
Summary
The paper introduces a new approach to establish central limit theorems (CLT) for functionals on binomial point processes and Poisson point processes. The random variable Nn has Poisson distribution with parameter n and is independent of {Xi} By a functional, it means a real-valued measurable function H defined on all finite subsets in Rd. We will study CLTs for H(n1/dXn) and H(n1/dPn) as n tends to infinity, where aX = {ax : x ∈ X} for a ∈ R and X ⊂ Rd. Let us first introduce the results in [18]. (Some additional bounded moments conditions are needed.) Note that we impose the strong stabilization on homogeneous Poisson point processes only. This condition is very mild in the sense that it is a sufficient condition for the well-established de-Poissonization technique in [16, Section 2.5].
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