Stein's method for discrete Gibbs measures

Abstract

Stein's method provides a way of bounding the distance of a probability distribution to a target distribution μ. Here we develop Stein's method for the class of discrete Gibbs measures with a density eV, 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, Hoist 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]. © Institute of Mathematical Statistics, 2008.