The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem

Haoran Wang

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arXiv · 2609.27433

The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem

Haoran Wang

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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 $0 1$, 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 $0<q\leq 1$. Below order one, the proof combines the Hillion--Johnson transport inequality with an explicit nonlinear telescoping correction. The corrected local curvature reduces to a two-dimensional quadratic form. An exact Riccati identity, together with a one-sided zero-crossing argument, proves positivity of its determinant throughout the full range $0<q<1$.

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Haoran Wang. 2026-09-23. The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem. https://arxiv.org/abs/2609.27433

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