A criterion for the compound poisson distribution to be maximum entropy

The Poisson distribution is known to have maximal entropy among all distributions (on the nonnegative integers) within a natural class. Interestingly, straightforward attempts to generalize this result to general compound Poisson distributions fail because the analogous result is not true in general. However, we show that the compound Poisson does indeed have a natural maximum entropy characterization when the distributions under consideration are log-concave. This complements the recent development by the same authors of an information-theoretic foundation for compound Poisson approximation inequalities and limit theorems.