Central limit theorem in Rényi divergence for lattice random variables

2026-08-17

AI Summary

This study addresses the absence of a Central Limit Theorem (CLT) and the challenges associated with asymptotic expansions for Rényi divergence of lattice random variables. By introducing quantized Gaussian distributions and strict sub-Gaussian conditions, this work systematically analyzes divergence convergence in discrete settings. It establishes necessary and sufficient conditions for the convergence of Rényi divergence and successfully derives Edgeworth-type asymptotic expansions of arbitrary order. Consequently, this research constructs a comprehensive CLT theoretical framework for lattice variables, effectively bridging critical gaps in discrete probability approximation theory and providing a rigorous mathematical foundation for related statistical inference.

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).

Problem

• Rényi divergence
• Central limit theorem
• Lattice random variables
• Edgeworth expansion
• Sub-Gaussian condition

Innovation

• Rényi divergence
• Central limit theorem
• Lattice random variables
• Edgeworth expansion
• Strict sub-Gaussian condition