Refined Cramér-type moderate deviations by Stein's method under dependence

Abstract

In this paper, we establish refined Cramér-type moderate deviations for dependent random variables at any fixed order $m\geq3$. Combining higher-order Stein identities with exponential tilting, we obtain cumulant-corrected tail approximations with explicit error bounds for boundedly supported kernels. Our main application gives a fourth-order expansion under bounded local dependence, reaching an $O(n^{3/10})$ deviation range under standard normalization with fixed dependence parameters. It improves on previous work in both deviation range and applicability, and also covers bounded $m$-dependent sequences without an independent-base representation. Further applications to the generalized combinatorial central limit theorem and the Curie-Weiss model also extend some earlier results under permutation and mean-field dependence.