Central limit theorem in Rényi divergence for lattice random variables
Abstract
Let {Xi}i=1n be independent, identically distributed lattice random variables with zero mean, unit variance, and maximal span h>0. Let Sn=(X1+⋯+Xn)/√n. Let Zn denote the quantization of a standard Gaussian random variable on the support lattice of Sn. We show that the Rényi divergence Dα(Sn|Zn) of order 1<α<∞ tends to 0 as n→∞ if and only if Dα(Sn0| Zn0)<∞ for some integer n0 and a sub-Gaussian condition holds. Furthermore, we derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. We also show that the Rényi divergence D∞(Sn|Zn) of infinite order tends to 0 as n→∞ if and only if a sub-Gaussian condition, a curvature condition, and a uniform separation condition hold. These results constitute lattice counterparts of Rényi entropic central limit theorems for continuous random variables obtained by Bobkov, Chistyakov and Götze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323) and Bobkov and Götze (\emph{Ann. Probab.} \textbf{53} (2025), 453--477). Uniform separation is an additional assumption beyond the sub-Gaussian and curvature conditions used in the continuous setting. The failure of uniform separation may cause an exponentially tilted version of X1 to concentrate on a sublattice of max span kh for some integer k>1, which yields the lower bound D∞(Sn|Zn)≥log k, and hence a failure of the entropic central limit theorem.
Related
- Optimal central limit theorem for bounded random variables in high dimensions
- A Central Limit Theorem For Random Variables Under A Constraint
- A central limit theorem for the degree of a random product of Cremona transformations
- Quenched functional central limit theorem for the random conductance model under minimal moments
- Random Hamiltonians II: A central limit theorem and the Hofer geometry of random walks
BibTeX
@misc{fu2026central,
title = {{Central limit theorem in Rényi divergence for lattice random variables}},
author = {Zhen Fu and Jiange Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.16144},
eprint = {2608.16144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16144},
note = {CC0},
}
RIS
TY - UNPB
TI - Central limit theorem in Rényi divergence for lattice random variables
AU - Zhen Fu
AU - Jiange Li
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.16144
UR - https://arxiv.org/abs/2608.16144
AB - Let {X_i}_i=1^n be independent, identically distributed lattice random variables with zero mean, unit variance, and maximal span h>0. Let S_n=(X_1+⋯+X_n)/√n. Let Z_n denote the quantization of a standard Gaussian random variable on the support lattice of S_n. We show that the Rényi divergence D_α(S_n|Z_n) of order 1<α<∞ tends to 0 as n→∞ if and only if D_α(S_n_0| Z_n_0)<∞ for some integer n_0 and a sub-Gaussian condition holds. Furthermore, we derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. We also show that the Rényi divergence D_∞(S_n|Z_n) of infinite order tends to 0 as n→∞ if and only if a sub-Gaussian condition, a curvature condition, and a uniform separation condition hold. These results constitute lattice counterparts of Rényi entropic central limit theorems for continuous random variables obtained by Bobkov, Chistyakov and Götze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323) and Bobkov and Götze (\emph{Ann. Probab.} \textbf{53} (2025), 453--477). Uniform separation is an additional assumption beyond the sub-Gaussian and curvature conditions used in the continuous setting. The failure of uniform separation may cause an exponentially tilted version of X_1 to concentrate on a sublattice of max span kh for some integer k>1, which yields the lower bound D_∞(S_n|Z_n)≥log k, and hence a failure of the entropic central limit theorem.
ER -