Central limit theorem in Rényi divergence for lattice random variables

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 $\alpha >1$, with $\beta =\alpha /(\alpha -1)$, we prove that the Rényi divergence $D_{\alpha }(S_n\Vert Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$\mathbb{E}{e}^{tX}<e^{\beta t^2/2},t\in \mathbb{R},t\ne 0$$ 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).