The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem
Abstract
Let B1,…,Bn be independent Bernoulli random variables with parameters p1,…,pn, and let S=∑i Bi. 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<q<1, the power sum ∑k ℙ(S=k)q is jointly concave in (p1,…,pn), and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order q are jointly concave. At q=1 this agrees with the Shannon theorem. For every q>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≤ 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.
Related
- Rényi and Tsallis information entropies for a harmonic position-dependent mass
- Uniqueness and nonuniqueness in the disjoint data inverse problem: a sharp spectral threshold
- Sharp Rogers--Shephard Inequalities for Rényi Entropy
- Rényi stability of Bh sets: a two-order phase diagram and sharp deletion principles
- The Sharp Ellipticity Threshold for W1,1-Regularity
BibTeX
@misc{wang2026sharp,
title = {{The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem}},
author = {Haoran Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27433},
eprint = {2609.27433},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27433},
}
RIS
TY - UNPB
TI - The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem
AU - Haoran Wang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27433
UR - https://arxiv.org/abs/2609.27433
AB - Let B_1,…,B_n be independent Bernoulli random variables with parameters p_1,…,p_n, and let S=∑_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<q<1, the power sum ∑_k ℙ(S=k)^q is jointly concave in (p_1,…,p_n), and strictly concave on the open parameter cube. Consequently, the R'enyi and Tsallis entropies of order q are jointly concave. At q=1 this agrees with the Shannon theorem. For every q>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≤ 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.
ER -