Stein's method for dependent random variables occurring in Statistical Mechanics

Abstract

We obtain rates of convergence in limit theorems of partial sums $S_n$ 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. Under appropriate assumptions there exists a real number $\alpha$, a positive real number $\mu$, and a positive integer $k$ such that $(S_n-n\alpha )/n^{1-1/2k}$ converges weakly to a random variable with density proportional to $\exp (-\mu |x{|}^{2k}/(2k)!)$. 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 with density proportional to $\exp (-\mu |x{|}^{2k}/(2k)!)$. Our results include the optimal 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 ${\beta }_c=1$, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature $1/{\beta }_n$ 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.

References in corpus (1)

• Stein's method for discrete Gibbs measures

Cited by in corpus (7)

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• Limit theorems for monomer-dimer mean-field models with attractive potential
• Berry-Esseen bounds in the inhomogeneous Curie-Weiss model with external field