Gaussian approximation for functionals of Gibbs particle processes

TL;DR

This paper extends point process theory techniques to the space of compact sets to analyze the asymptotic Gaussian behavior of functionals of Gibbs particle processes, particularly in planar Gibbs segment processes.

Contribution

It adapts and applies the Malliavin-Stein calculus and existence conditions for Gibbs processes to the space of compact sets, enabling Gaussian approximation results.

Findings

Derived asymptotic Gaussian approximation for functionals of Gibbs segment processes.

Extended point process techniques to the space of compact sets with Hausdorff metric.

Simplified conditions for the existence of stationary Gibbs processes.

Abstract

In the paper asymptotic properties of functionals of stationary Gibbs particle processes are derived. Two known techniques from the point process theory in the Euclidean space R^d are extended to the space of compact sets on R^d equipped by the Hausdorff metric. First, conditions for the existence of the stationary Gibbs point process with given conditional intensity have been simplified recently. Secondly, the Malliavin-Stein calculus was applied to the estimation of Wasserstein distance between the Gibbs input and standard Gaussian distribution. We transform these theories to the space of compact sets and use them to derive a Gaussian approximation for functionals of a planar Gibbs segment process.