Central limit theorem in R\'enyi divergence for lattice random variables

Abstract

We establish a central limit theorem in R\'enyi 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 $\alpha>1$, with $\beta=\alpha/(\alpha-1)$, we prove that the R\'enyi divergence $D_\alpha(S_n\|Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$ \mathbb E e^{tX}<e^{\beta t^2/2},\quad t\in\mathbb R,~ t\ne0 $$ 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 R\'enyi entropic central limit theorem for continuous random variables due to Bobkov, Chisyakov and G\"{o}tze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323).