NONNORMAL APPROXIMATION BY STEIN'S METHOD OF EXCHANGEABLE PAIRS WITH APPLICATION TO THE CURIE-WEISS MODEL

• 成果类型: Article
• 署名作者: Chatterjee, Sourav; Shao, Qi-Man
• 署名单位: New York University; Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP712
• 发表日期: 2011
• 页码: 464-483
• 关键词: limit-theorems; sums

摘要

where g(W) is a dominated term and r(W) is negligible. Let G(t) = f(0)(t)g(s)ds and define p(t) = c1e(-c0G(t)), where c(0) is a properly chosen constant and c(1) = 1/integral(infinity)(-infinity) 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 1/root 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.