On the Maximum Entropy of a Sum of Independent Discrete Random Variables

TL;DR

This paper investigates the maximum possible entropy of the sum of independent discrete variables over finite alphabets, extending known binary results to larger alphabets and proposing a conjecture for the optimal distribution.

Contribution

It generalizes the Shepp--Olkin theorem to arbitrary finite alphabets, providing a lower bound on the maximum entropy of the sum and supporting a conjecture on the optimal distribution structure.

Findings

Lower bound on the maximum entropy of the sum for arbitrary alphabets

The bound is tight in several special cases

Conjecture that the maximum is achieved with specific mixture distributions

Abstract

Let $ X_1, \ldots, X_n $ be independent random variables taking values in the alphabet $ \{0, 1, \ldots, r\} $, and $ 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, \ldots, X_{n-1} $ are uniformly distributed over $ \{0, r\} $, while the probability mass function of $ X_n $ is a mixture (with explicitly defined non-zero weights) of the uniform distributions over $ \{0, r\} $ and $ \{1, \ldots, r-1\} $.