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≥3. 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(n3/10) 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.
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BibTeX
@misc{liu2026refined,
title = {{Refined Cramér-type moderate deviations by Stein's method under dependence}},
author = {Hui-Lin Liu and Song-Hao Liu and Dawei Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29078},
eprint = {2610.29078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29078},
}
RIS
TY - UNPB
TI - Refined Cramér-type moderate deviations by Stein's method under dependence
AU - Hui-Lin Liu
AU - Song-Hao Liu
AU - Dawei Lu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29078
UR - https://arxiv.org/abs/2610.29078
AB - In this paper, we establish refined Cramér-type moderate deviations for dependent random variables at any fixed order m≥3. 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.
ER -