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 $G(t)=\int_{0}^{t}g(s)ds$ and define $p(t)=c_{1}e^{-c_{0}G(t)}$ , where c₀ is a properly chosen constant and $c_{1}=1/\int_{-\infty}^{\infty }e^{-c_{0}G(t)}dt$ . 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.