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.
Highlights
Stein’s method is a powerful tool to prove distributional approximation
The aim of this paper is to develop Stein’s method for exchangeable pairs for a rich class of distributional approximations and thereby prove Berry-Esseen bounds for the sums of dependent random variables occurring in statistical mechanics under the name Curie-Weiss models
Approximations with fixed limiting laws can be obtained in the classical case, since we can apply Corollary 2.9 and part (2) of Theorem 2.4: Theorem 3.7
Summary
Stein’s method is a powerful tool to prove distributional approximation. One of its advantages is that is often automatically provides a rate of convergence and may be applied rather effectively to classes of random variables that are stochastically dependent. The aim of this paper is to develop Stein’s method for exchangeable pairs (see [20]) for a rich class of distributional approximations and thereby prove Berry-Esseen bounds for the sums of dependent random variables occurring in statistical mechanics under the name Curie-Weiss models. This inequality ensures the application of correlation-inequalities due to Lebowitz for bounding the variances and other low order moments, which appear in Stein’s method. As far as we understand, there the authors give an alternative proof of Theorem 3.7 and 3.8
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