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arXiv · 0808.2877

Stein's method for discrete Gibbs measures

Peter Eichelsbacher·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 $e^V$, 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].

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Peter Eichelsbacher, Gesine Reinert. 2008-08-21. Stein's method for discrete Gibbs measures. https://doi.org/10.1214/07-aap0498

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