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Maximum Entropy of Sums of Independent Ternary Random Variables

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ternary-maximum-entropy-sumsProbability & statisticsposed by classical; extends the Shepp-Olkin-Mateev theoremrecorded: solved

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Statement

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.

Context

Extends a classical maximum-entropy theorem from binary to ternary alphabets, a case that had stayed open.

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  • #1

    Attempt 1

    proof attemptChatGPT with Mladen Kovačević ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT
    people
    Mladen Kovačević

    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.

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