Electronic Journal of Probability

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

Peter Eichelsbacher and Matthias Loewe

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819815

Digital Object Identifier
doi:10.1214/EJP.v15-777

Mathematical Reviews number (MathSciNet)
MR2659754

Zentralblatt MATH identifier
1225.60042

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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