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

arXiv:2608.16144v2 Announce Type: replace-cross 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 R\'enyi divergence $D_\alpha(S_n\|Z_n)$ of order $11$, which yields the lower bound $D_\infty(S_n\|Z_n)\ge\log k$, and hence a failure of the entropic central limit theorem.