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 is considered and settled in the ternary case. Namely, the following theorem is established: if $X_1,\dots ,X_n$ are independent random variables taking values in $\{0,1,2\}$, then the entropy of $S_n=X_1+\cdots +X_n$ is maximized when $X_1,\dots ,X_{n-1}$ are uniform on $\{0,2\}$ and the probability mass function of $X_n$ is given by $\text{Prob}(X_n=0)=\text{Prob}(X_n=2)=w\mathrm{/}2$, $\text{Prob}(X_n=1)=1-w$, where $w=(1+2^{-H(B_n)+H(B_{n-1})}{)}^{-1}$ and $B_m\sim \text{Bin}(m,1\mathrm{/}2)$. 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.