Discoveries / probability-statistics

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 $X_1, \ldots, X_n$ taking values in $\{0,1,2\}$, 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 $X_n$ has an explicitly described three-point distribution. This extends the Shepp-Olkin-Mateev theorem to ternary alphabets.