Electronic Journal of Probability

Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics

Peter Eichelsbacher and Matthias Loewe

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We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions. As a consequence we obtain convergence rates in limit theorems of partial sums for certain sequences of dependent, identically distributed random variables which arise naturally in statistical mechanics, in particular in the context of the Curie-Weiss models. Our results include a Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 30, 962-988.

Accepted: 28 June 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general)

Berry-Esseen bound Stein's method exchangeable pairs Curie Weiss models critical temperature GHS-inequality

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Eichelsbacher, Peter; Loewe, Matthias. Stein's Method for Dependent Random Variables Occuring in Statistical Mechanics. Electron. J. Probab. 15 (2010), paper no. 30, 962--988. doi:10.1214/EJP.v15-777. https://projecteuclid.org/euclid.ejp/1464819815

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