  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\} $.
