Let B_1,…,B_n be independent Bernoulli random variables with parameters p_1,…,p_n∈[0,1], let S=∑_i=1^n B_i, and write P_k(p)=P_p(S=k). Determine the orders q>0 for which, for every n, the Rényi entropy H_q^R(S)=frac11-qlog!(∑_k P_k(p)^q) and the Tsallis entropy H_q^T(S)=frac11-q(∑_k P_k(p)^q-1) are jointly concave functions of p=(p_1,…,p_n). The claimed sharp answer is that both entropies are jointly concave for every 0 1, already when n=2.

Hillion–Johnson critical-order conjectures for the Shepp–Olkin concavity problem

Let $B_1,\ldots,B_n$ be independent Bernoulli random variables with parameters $p_1,\ldots,p_n\in[0,1]$ , let $S=\sum_{i=1}^n B_i$ , and write $P_k(p)=\mathbb{P}_p(S=k)$ . Determine the orders $q>0$ for which, for every $n$ , the Rényi entropy $H_q^{\mathrm{R}}(S)=\frac{1}{1-q}\log\!\left(\sum_k P_k(p)^q\right)$ and the Tsallis entropy $H_q^{\mathrm{T}}(S)=\frac{1}{1-q}\left(\sum_k P_k(p)^q-1\right)$ are jointly concave functions of $p=(p_1,\ldots,p_n)$ . The claimed sharp answer is that both entropies are jointly concave for every $0<q\leq 1$ , while joint concavity fails for every $q>1$ , already when $n=2$ .

References

Primary source

arXiv

Additional references

The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem — arXiv — Haoran Wang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A new unrefereed preprint claims to settle the universal concavity ranges for two generalized entropy families, overturning earlier predictions.

Hillion and Johnson proposed critical-order conjectures for Rényi and Tsallis entropies in their Shepp–Olkin work, following the 1981 Shannon-entropy conjecture of Shepp and Olkin. The conjectures predicted thresholds $q_R^*=2$ and $q_T^*=3.65986\ldots$ , with concavity below each threshold.

Known results

The Shannon Shepp–Olkin concavity conjecture was proved by Hillion and Johnson (2017).
The monotone-parameter case was proved earlier (Hillion and Johnson, 2013).
Hillion and Johnson established failure above $q_R^*=2$ for Rényi entropy.
They established failure above $q_T^*=3.65986\ldots$ , the root of $2-4q+2^q=0$ , for Tsallis entropy.

September 2026 claimed exact range

On September 23, 2026, Haoran Wang's preprint The Sharp Rényi and Tsallis Threshold in the Shepp--Olkin Concavity Problem claimed joint concavity below order one and a two-variable obstruction above order one, contradicting the earlier predicted thresholds. The claim is supported only by an unrefereed preprint.

Current status (as of September 2026): A preprint claims the Rényi and Tsallis ranges are exactly determined, but this resolution remains unverified.

Sources

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