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...

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Bibliographic Details
Published in:The Annals of applied probability 2011-04, Vol.21 (2), p.464-483
Main Authors: Chatterjee, Sourav, Shao, Qi-Man
Format: Article
Language:English
Subjects:
60F05
60G09
Approximation
Berry–Esseen bound
Critical temperature
Curie–Weiss model
Error bounds
exchangeable pair
Laplace transforms
Magnetization
Markov analysis
Markov chains
Mathematical theorems
Probability distribution
Random variables
Statistical mechanics
Statistics
Stein’s method
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Summary: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.
ISSN:1050-5164
2168-8737
DOI:10.1214/10-AAP712