On the Maximum Entropy of a Sum of Independent Discrete Random Variables
Mladen Kovačević · Theory of Probability and Its Applications · 2021
Let $X_1, \dots, X_n$ be independent random variables taking values in the alphabet $\{0, 1, \dots, r\}$, and let $S_n=\sum_{i=1}^n X_i$. The Shepp--Olkin theorem states that in the binary case (${r=1}$), the Shannon entropy of $S_n$ is maximized when all the $X_i$'s are uniformly distributed, i.e., Bernoulli(1/2). In an attempt to generalize this theorem to arbitrary finite alphabets, we obtain a lower bound on the maximum entropy of $S_n$ and prove that it is tight in several special cases. In addition to these special cases, an argument is presented supporting the conjecture that the bound represents the optimal value for all $n$, $r$, i.e., that $H(S_n)$ is maximized when $X_1, \dots, X_{n-1}$ are uniformly distributed over $\{0, r\}$, while the probability mass function of $X_n$ is a mixture (with explicitly defined nonzero weights) of the uniform distributions over $\{0, r\}$ and $\{1, \dots, r-1\}$.