Year: 2026
Type: preprint
Abstract: Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n$, and let $S=\sum_i B_i$. Hillion and Johnson proved that the Shannon entropy of $S$ is jointly concave in the parameter vector and proposed corresponding critical-order conjectures for R'enyi and Tsallis entropies, with predicted thresholds $2$ and approximately $3.65986$, respectively. We determine both thresholds exactly. For every $01$, joint concavity fails already for the sum of two Bernoulli variables: a transverse interpolation in which the two parameters move in opposite directions gives strict local convexity for both entropies. Hence the universal joint-concavity range for both families is exactly $0
Source: arXiv (Cornell University)
Author: Wang, Haoran
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214171386