Nonnormal approximation by stein's method of exchangeable pairs with application to the curie-weiss model

Let (W,W¹) be an exchangeable pair. Assume that mathmatical iqradic represented, where g(W) is a dominated term and r(W) is negligible. Let mathmatical iqradic represented ds and define mathmatical represented, where c0 is a properly chosen constant and mathmatical iqradic represented 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 ¹) 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.