The Annals of Applied Probability

Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model

Sourav Chatterjee and Qi-Man Shao

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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.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 2 (2011), 464-483.

Dates
First available in Project Euclid: 22 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1300800979

Digital Object Identifier
doi:10.1214/10-AAP712

Mathematical Reviews number (MathSciNet)
MR2807964

Zentralblatt MATH identifier
1216.60018

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G09: Exchangeability

Keywords
Stein’s method exchangeable pair Berry–Esseen bound Curie–Weiss model

Citation

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


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