Central limit theorem in Rényi divergence for lattice random variables

Abstract

Let $\{X_i{\}}_{i=1}^n$ be independent, identically distributed lattice random variables with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots +X_n)/\sqrt{n}$. 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 $D_{\alpha }(S_n\Vert Z_n)$ of order $1<\alpha <\infty$ tends to 0 as $n\to \infty$ if and only if $D_{\alpha }(S_{n_0}\Vert Z_{n_0})<\infty$ 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 $D_{\infty }(S_n\Vert Z_n)$ of infinite order tends to 0 as $n\to \infty$ 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 $D_{\infty }(S_n\Vert Z_n)\ge \log k$, and hence a failure of the entropic central limit theorem.

arXiv commentAdded the infinite order Rényi entropic CLT