Stein’s method for discrete Gibbs measures
Peter Eichelsbacher and Gesine Reinert
Source: Ann. Appl. Probab.
Volume 18, Number 4
(2008), 1588-1618.
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].
Primary Subjects: 60E05
Secondary Subjects: 60F05, 60E15, 82B05
Keywords: Stein’s method; Gibbs measures; birth and death processes; size bias coupling
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1216677133
Digital Object Identifier: doi:10.1214/07-AAP0498
Mathematical Reviews number (MathSciNet):
MR2434182
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