From Stein identities to moderate deviations

Abstract

Stein's method is applied to obtain a general Cramer-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie-Weiss model. A general moderate deviation result for independent random variables is also proved.

arXiv commentPublished in at http://dx.doi.org/10.1214/12-AOP746 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Normal approximation under local dependence
• A new method of normal approximation
• Stein's method and the zero bias transformation with application to simple random sampling
• Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model
• Applications of Stein's method for concentration inequalities
• Berry Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

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• Limit Theorems in the Imitative Monomer-Dimer Mean-Field Model via Stein's Method
• On moderate deviations in Poisson approximation