Maximum Entropy of Sums of Independent Ternary Random Variables
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911673670959104 |
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| author | Kovačević, Mladen |
| author_facet | Kovačević, Mladen |
| contents | The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if \(X_1,\ldots,X_n\) are independent random variables taking values in \(\{0,1,2\}\), then the entropy of \(S_n=X_1+\cdots+X_n\) is maximized when \(X_1,\ldots,X_{n-1}\) are uniform on \(\{0,2\}\) and the probability mass function of \(X_n\) is given by \(\Prob(X_n=0) = \Prob(X_n=2) = w/2\), \(\Prob(X_n=1) = 1-w\), where \(w = \big(1 + 2^{-H(B_n)+H(B_{n-1})}\big)^{-1}\) and \(B_m\sim \Bin(m,1/2)\). The statement can be seen as an extension to ternary alphabets of the Shepp--Olkin--Mateev theorem. The proof uses the Hermite--Biehler theorem, Newton's inequalities, and Yu's maximum-entropy theorem for ultra-log-concave distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11831 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximum Entropy of Sums of Independent Ternary Random Variables Kovačević, Mladen Information Theory Discrete Mathematics Probability 94A17, 60C05, 60G50 The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if \(X_1,\ldots,X_n\) are independent random variables taking values in \(\{0,1,2\}\), then the entropy of \(S_n=X_1+\cdots+X_n\) is maximized when \(X_1,\ldots,X_{n-1}\) are uniform on \(\{0,2\}\) and the probability mass function of \(X_n\) is given by \(\Prob(X_n=0) = \Prob(X_n=2) = w/2\), \(\Prob(X_n=1) = 1-w\), where \(w = \big(1 + 2^{-H(B_n)+H(B_{n-1})}\big)^{-1}\) and \(B_m\sim \Bin(m,1/2)\). The statement can be seen as an extension to ternary alphabets of the Shepp--Olkin--Mateev theorem. The proof uses the Hermite--Biehler theorem, Newton's inequalities, and Yu's maximum-entropy theorem for ultra-log-concave distributions. |
| title | Maximum Entropy of Sums of Independent Ternary Random Variables |
| topic | Information Theory Discrete Mathematics Probability 94A17, 60C05, 60G50 |
| url | https://arxiv.org/abs/2605.11831 |