The Annals of Applied Probability

Stein’s method for discrete Gibbs measures

Peter Eichelsbacher and Gesine Reinert

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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, 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].

Article information

Source
Ann. Appl. Probab., Volume 18, Number 4 (2008), 1588-1618.

Dates
First available in Project Euclid: 21 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1216677133

Digital Object Identifier
doi:10.1214/07-AAP0498

Mathematical Reviews number (MathSciNet)
MR2434182

Zentralblatt MATH identifier
1146.60011

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60F05: Central limit and other weak theorems 60E15: Inequalities; stochastic orderings 82B05: Classical equilibrium statistical mechanics (general)

Keywords
Stein’s method Gibbs measures birth and death processes size bias coupling

Citation

Eichelsbacher, Peter; Reinert, Gesine. Stein’s method for discrete Gibbs measures. Ann. Appl. Probab. 18 (2008), no. 4, 1588--1618. doi:10.1214/07-AAP0498. https://projecteuclid.org/euclid.aoap/1216677133


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