Stein's method for discrete Gibbs measures

• 成果类型: Article
• 署名作者: Eichelsbacher, Peter; Reinert, Gesine
• 署名单位: Ruhr University Bochum; University of Oxford
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP0498
• 发表日期: 2008
• 页码: 1588-1618
• 关键词: convergence

摘要

Stein's method provides a way of bounding the distance of a probability distribution to a target distribution jL. 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, 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.