A reverse entropy power inequality for i.i.d. log-concave random variables
Zhen Fu, Jiange Li
TL;DR
The paper establishes reverse entropy power inequalities for i.i.d. log-concave random variables, showing that the infinity-order Rényi entropy of the sum is maximized by exponential components, i.e., $h_\infty(X+Y)\le h_\infty(Z+W)$ with $Z,W$ exponential and $h_\infty(Z)=h_\infty(X)$, $h_\infty(W)=h_\infty(Y)$. It develops a toolkit based on decreasing rearrangement and majorization, along with change-of-measure techniques, to compare densities and their sums, and extends results to $p$-order Rényi entropies (all $0<p\le\infty$) in the i.i.d. setting. The paper also provides discrete analogs for integer-valued monotone log-concave variables, proving sharp bounds like $H_2(X+Y)<H_2(X)+\log 2$ and $H_\infty(X+Y)<H_\infty(X)+1$, with tightness demonstrated via geometric distributions. Overall, it links reverse EPI phenomena to Rogozin-type results and convex-analytic majorization, enriching the connection between entropy, rearrangements, and geometric inequalities in both continuous and discrete settings.
Abstract
We show that $h_\infty(X+Y)\leq h_\infty(Z+W)$, where $X, Y$ are independent log-concave random variables, and $Z, W$ are exponential random variables having the same respective $\infty$-Rényi entropies. Analogs for integer-valued monotone log-concave random variables are also obtained. Our main tools are decreasing rearrangement, majorization, and the change of measure.