Normal approximation for Gibbs processes via disagreement couplings

Electronic Journal of Probability, vol. 30, pp. 1–56

Abstract

This work improves the existing central limit theorems (CLTs) for geometric functionals of Gibbs processes in three aspects. First, we derive a CLT for weakly stabilizing functionals, thereby improving on the previously used assumption of exponential stabilization. Second, we show that this CLT holds for interaction ranges up to the percolation threshold of the dominating Poisson process. This avoids imprecise branching bounds from graphical construction. Third, by constructing simultaneous couplings of several Palm processes for Gibbs functionals, we provide a quantitative CLT in terms of Kolmogorov bounds for normal approximation. An important conceptual ingredient in these advances is the extension of disagreement coupling adapted to increasing windows and to the comparison at multiple spatial locations.

Authors 3

1. Christian Hirsch

   Aarhus University

   Affiliation as printed

   DIGIT Center and Department of Mathematics, Aarhus University

2. Moritz Otto Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University

3. Anne Marie Svane

   Aalborg University

   Affiliation as printed

   Department of Mathematics, Aalborg University