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/.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.