PreprintCC BY 4.0arXiv
A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem
Abstract
We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let (ξi,ηi)i≥ 1 be independent copies of a pair (ξ,η), and set Wn=n−1/2∑i=1n ξi and ηn=(η1,…,ηn). Under the assumptions that 𝔼Var(ξ|η)<∞ and that the conditional law of ξ given η is absolutely continuous almost surely, we show that 𝔼h(Wn|ηn) converges to the Gaussian entropy 1/2log(2πeσ2), where σ2=𝔼Var(ξ|η), if and only if 𝔼h(Wn0|ηn0)>−∞ for some n0. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.
Related
- Improved Bounds for the Entropic Central Limit Theorem in the Log-Concave Case
- Identified strain kernel A(θ,ω) and a conditional regularity criterion for 3D Navier–Stokes (κ = 3/2; working notes)
- Proof of the entropic c-theorem
- A conditional no-splash criterion for two-fluid Euler--MHD interface
- Entropy-Regularized Mean-Field Games with Average-Reward Criterion: Span-Seminorm Contractions and Model-Free Learning
BibTeX
@misc{ye2026finiteentropy,
title = {{A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem}},
author = {Tong Ye and Liu-Quan Yao and Shuai Yuan and Guanghui Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30579},
eprint = {2608.30579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30579},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem
AU - Tong Ye
AU - Liu-Quan Yao
AU - Shuai Yuan
AU - Guanghui Wang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30579
UR - https://arxiv.org/abs/2608.30579
AB - We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let (ξ_i,η_i)_i≥ 1 be independent copies of a pair (ξ,η), and set W_n=n^−1/2∑_i=1^n ξ_i and η_n=(η_1,…,η_n). Under the assumptions that 𝔼Var(ξ|η)<∞ and that the conditional law of ξ given η is absolutely continuous almost surely, we show that 𝔼h(W_n|η_n) converges to the Gaussian entropy 1/2log(2πeσ^2), where σ^2=𝔼Var(ξ|η), if and only if 𝔼h(W_n_0|η_n_0)>−∞ for some n_0. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.
ER -