Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

Let (W, W') be an exchangeable pair. Assume that E(W − W'|W) = g(W) + r(W), where g(W) is a dominated term and r(W) is negligible. Let G(t) = ∫0tg(s) ds and define p(t) = c1e−c0G(t), where c0 is a properly chosen constant and c1 = 1 / ∫−∞∞e−c0G(t) dt. Let Y be a random variable with the probability density function p. It is proved that W converges to Y in distribution when the conditional second moment of (W − W') given W satisfies a law of large numbers. A Berry–Esseen type bound is also given. We use this technique to obtain a Berry–Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie–Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli–Laplace Markov chain is also discussed.