From Stein identities to moderate deviations

TL;DR

This paper extends Stein's method to derive moderate deviation results for dependent and independent random variables, with applications to various probabilistic models and combinatorial structures.

Contribution

It introduces a general framework for moderate deviations based on Stein identities, including new results for dependent variables and specific models.

Findings

Established a general Cramer-type moderate deviation theorem for dependent variables.

Derived moderate deviation results for independent variables.

Applied the theory to models like the Curie-Weiss and anti-voter models.

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.