## The Annals of Applied Probability

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

#### 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

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

#### 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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