Central limit theorem in Rényi divergence for lattice random variables
2026-08-17 • Information Theory
Information Theory
AI summaryⓘ
The authors study how sums of certain types of discrete random variables behave when scaled and compared to a standard normal distribution on matching lattice points. They prove that a specific measure of difference called Rényi divergence goes to zero under precise conditions involving how the variables' distributions behave, especially a 'sub-Gaussian' constraint on their moment generating functions. Additionally, they provide detailed expansions describing how fast this convergence happens. This work extends existing central limit results for continuous variables to the lattice (discrete) setting.
Central Limit TheoremRényi DivergenceLattice Random VariablesSub-Gaussian ConditionEdgeworth ExpansionMoment Generating FunctionConvolutionQuantizationStandard Gaussian Distribution
Authors
Zhen Fu, Jiange Li
Abstract
We establish a central limit theorem in Rényi divergence for independent and identically distributed lattice random variables $X_1, \cdots, X_n$ with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the standard Gaussian distribution quantized on the support lattice of $S_n$. For every $α>1$, with $β=α/(α-1)$, we prove that the Rényi divergence $D_α(S_n\|Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$ \mathbb E e^{tX}<e^{βt^2/2},\quad t\in\mathbb R,~ t\ne0 $$ holds. Under these conditions, we further derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. These results provide a lattice counterpart of the Rényi entropic central limit theorem for continuous random variables due to Bobkov, Chisyakov and Götze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323).