Bounds for the probability generating functional of a Gibbs point process

TL;DR

This paper establishes explicit bounds for the probability generating functional of stationary Gibbs point processes, enabling improved analysis of summary statistics and correlation functions using Stein's method.

Contribution

It introduces novel bounds for the probability generating functional of Gibbs point processes, enhancing the analysis of their statistical properties.

Findings

Derived explicit bounds for the probability generating functional.

Provided estimates for G and K functions, intensity, and correlation functions.

Applied Stein's method for Poisson approximation to Gibbs processes.

Abstract

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics like the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity and higher order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.