We obtain rates of convergence in limit theorems of partial sums Sn 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
α, a positive real number μ, and a positive integer k such that
(Sn−nα)/n1−1/2k converges weakly to a random variable with
density proportional to exp(−μ∣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(−μ∣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 βc=1, where the
Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the
temperature 1/βn 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