Stein's method for discrete Gibbs measures

Details Stein's method for discrete Gibbs measures Stein's method for discrete Gibbs measures

2008-08-21

arxiv.org/abs/0808.2877v1

Annals of Applied Probability 2008, Vol. 18, No. 4, 1588-1618

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP0498 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Ins

Scientific paper

10.1214/07-AAP0498

Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $\mu$. 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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