TITLE: Infinitely Divisible Distributions and Maximal Entropy
QUESTION [5 upvotes]: The normal distribution on $\mathbb{R}$, the exponential distribution on $\mathbb{R}_{\geq 0}$, and the geometric distribution on $\mathbb{N}$ are examples of distributions that are both infinitely divisible and entropy maximizers.  On the other hand, the Poisson distribution is an infinitely divisible distribution on $\mathbb{N}$ without maximizing entropy, while the uniform distribution on the interval $[a,b]$ maximizes entropy but is not infinitely divisible.
Can anything be said about the relationship between these two classes of distributions?

REPLY [5 votes]: The Poisson distribution has maximum entropy under the condition of having a specific mean and being a sum of Bernoulli random variables. The condition of being a sum of Bernoulli random variables can be weakened to being ultra log-concave.