Stein's method for discrete Gibbs measures

TL;DR

This paper extends Stein's method to discrete Gibbs measures, providing new bounds and approximation techniques for Gibbs ensembles, Poisson, and geometric distributions, with applications to strongly correlated variables.

Contribution

It develops Stein's method for discrete Gibbs measures using size bias couplings and improves existing bounds for Poisson and geometric approximations.

Findings

Derived bounds for Gibbs measure approximations.

Improved Poisson approximation bounds for sums of indicators.

Enhanced nonuniform Stein bounds for geometric distributions.

Abstract

Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $\mu$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) 30--42]. We obtain estimates of the approximation to a grand-canonical Gibbs ensemble. As side results, we slightly improve on the Barbour, Holst and Janson [Poisson Approximation (1992)] bounds for Poisson approximation to the sum of independent indicators, and in the case of the geometric distribution we derive better nonuniform Stein bounds than Brown and Xia [Ann. Probab. 29 (2001) 1373--1403].