Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

TL;DR

This paper investigates the maximum entropy properties of compound Poisson and related distributions, establishing conditions for log-concavity and extending previous results to provide an information-theoretic foundation for approximation theorems.

Contribution

It extends the semigroup approach to show maximum entropy properties for compound Poisson distributions under log-concavity, and explores applications to combinatorics and entropy bounds.

Findings

Compound Poisson has maximum entropy under log-concavity.

Established maximum entropy for compound binomial measures.

Derived new entropy bounds for combinatorial structures.

Abstract

Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be log-concave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a claw-free graph, and a connection is drawn to Mason's conjecture for matroids. The present results are primarily motivated by the desire to provide an information-theoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of log-concavity and ultra-log-concavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.