The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 21, Number 2 (2011), 464-483.
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 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.
Ann. Appl. Probab., Volume 21, Number 2 (2011), 464-483.
First available in Project Euclid: 22 March 2011
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Chatterjee, Sourav; Shao, Qi-Man. Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 (2011), no. 2, 464--483. doi:10.1214/10-AAP712. https://projecteuclid.org/euclid.aoap/1300800979