PreprintCC BY 4.0arXiv
A Berry--Esseen Bound of Order (1+log n)/√n for Peng's Central Limit Theorem
Abstract
We derive a quantitative central limit theorem under sublinear expectations from a spatial endpoint regularity estimate for the one-dimensional G-heat equation. For bounded Lipschitz initial data, the second spatial derivative and the time derivative of its solution are Lipschitz in space and 1/2-Hölder in time away from the initial time. In particular, [uxx(t,·)]Lip≤ CGLip(ϕ)/t. Combining this estimate with the nonlinear Stein identity introduced in Song \cite{Song2020} and a replacement argument yields a bound of order (1+log n)/√n for bounded Lipschitz tests when the third absolute moment is finite and the lower variance is positive. The final replacement step is estimated directly, avoiding the nonintegrable singularity at time zero.
Related
- The Central Limit Theorem and Berry--Esseen bound for logarithmic law of random determinants
- The Berry--Esseen estimate in the Boolean central limit theorem
- The multivariate Berry-Esseen theorem with an absolute constant
- The Berry-Esseen Constant Conjecture is Eventually True
- Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
BibTeX
@misc{song2026berryesseen,
title = {{A Berry--Esseen Bound of Order $(1+\log n)/\sqrt n$ for Peng's Central Limit Theorem}},
author = {Yongsheng Song},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15419},
eprint = {2610.15419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15419},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Berry--Esseen Bound of Order (1+log n)/√n for Peng's Central Limit Theorem
AU - Yongsheng Song
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15419
UR - https://arxiv.org/abs/2610.15419
AB - We derive a quantitative central limit theorem under sublinear expectations from a spatial endpoint regularity estimate for the one-dimensional G-heat equation. For bounded Lipschitz initial data, the second spatial derivative and the time derivative of its solution are Lipschitz in space and 1/2-Hölder in time away from the initial time. In particular, [u_xx(t,·)]_Lip≤ C_GLip(ϕ)/t. Combining this estimate with the nonlinear Stein identity introduced in Song \cite{Song2020} and a replacement argument yields a bound of order (1+log n)/√n for bounded Lipschitz tests when the third absolute moment is finite and the lower variance is positive. The final replacement step is estimated directly, avoiding the nonintegrable singularity at time zero.
ER -