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April 2011 Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model
Sourav Chatterjee, Qi-Man Shao
Ann. Appl. Probab. 21(2): 464-483 (April 2011). DOI: 10.1214/10-AAP712

Abstract

Let (W, W') be an exchangeable pair. Assume that

E(WW'|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) = c1ec0G(t), where c0 is a properly chosen constant and c1 = 1 / −∞ec0G(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 (WW') 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.

Citation

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Sourav Chatterjee. Qi-Man Shao. "Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model." Ann. Appl. Probab. 21 (2) 464 - 483, April 2011. https://doi.org/10.1214/10-AAP712

Information

Published: April 2011
First available in Project Euclid: 22 March 2011

zbMATH: 1216.60018
MathSciNet: MR2807964
Digital Object Identifier: 10.1214/10-AAP712

Subjects:
Primary: 60F05
Secondary: 60G09

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.21 • No. 2 • April 2011
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