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

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.

30 pages. This submission supersedes arXiv:0805.4112v1. Changes in v2: Updated references, typos corrected

References in corpus (5)

• Log-concavity and LC-positivity
• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Compound Poisson Approximation via Information Functionals
• Preservation of log-concavity on summation
• Log-concavity of characteristic polynomials and the Bergman fan of matroids

Cited by in corpus (18)

• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Forward and Reverse Entropy Power Inequalities in Convex Geometry
• Compound Poisson Approximation via Information Functionals
• Rogozin's convolution inequality for locally compact groups
• Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam region
• Entropy Inequalities for Sums in Prime Cyclic Groups
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• An Information-Theoretic Perspective of the Poisson Approximation via the Chen-Stein Method
• Bernoulli sums and Rényi entropy inequalities
• Matroids and log-concavity
• Entropies of weighted sums in cyclic groups and an application to polar codes
• Reversals of Rényi Entropy Inequalities under Log-Concavity
• Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$
• Entropy-variance inequalities for discrete log-concave random variables via degree of freedom
• Entropy and thinning of discrete random variables
• Efron's monotonicity property for measures on ${\mathbb{R}}^2$
• Tight Revenue Gaps among Multi-Unit Mechanisms
• Maximum Entropy of Sums of Independent Ternary Random Variables