Stein's method for dependent random variables occuring in statistical mechanics
Peter Eichelsbacher, Ruhr University of Bochum
Matthias Loewe, University of Muenster
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.
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