VibeMathedMath problems solved by AI

Maximum Entropy of Sums of Independent Ternary Random Variables

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet, settled in the ternary case. For independent X1,…,XnX_1, \ldots, X_n taking values in {0,1,2}\{0,1,2\}, the entropy of Sn=X1+⋯+XnS_n = X_1 + \cdots + X_n is maximized when X1,…,Xn−1X_1, \ldots, X_{n-1} are uniform on {0,2}\{0,2\} and XnX_n has an explicitly described three-point distribution. This extends the Shepp-Olkin-Mateev theorem to ternary alphabets.

Result
Proved
Status
Resolved
AI contribution
AI-assisted
Method
Argument
Field
Information theory
Posed by
classical; extends the Shepp-Olkin-Mateev theorem
Year posed
—
Years open
—
Solved
2026-05-12
Model
ChatGPT
Vendor
OpenAI
Collaborators
Mladen Kovačević
Verification
Unreviewed
Publication
Preprint
Significance
20 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What the AI did

At the weak end of what the catalog records. The author used ChatGPT to verify some of the derivations and to assist with formatting, reviewed and edited the content, and takes full responsibility for it. Verifying derivations is a mathematical use rather than a purely editorial one, which is why this is listed at all, but no idea in the paper is credited to the model.

Verification

arXiv preprint, not peer-reviewed. The proof runs through the Hermite-Biehler theorem, Newton's inequalities and Yu's maximum-entropy theorem for ultra-log-concave distributions, all standard tools, so it is checkable by a specialist.

Source

arXiv:2605.11831 - Maximum Entropy of Sums of Independent Ternary Random Variables

Discussion