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On the entropy and log-concavity of compound Poisson measures

Johnson, O and Kontoyiannis, I and Madiman, M On the entropy and log-concavity of compound Poisson measures. (Unpublished)

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Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation), this work examines sufficient conditions under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. We show that the natural analog of the Poisson maximum entropy property remains valid if the measures 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. The proofs are largely based on ideas related to the semigroup approach introduced in recent work by Johnson for the Poisson family. Sufficient conditions are given for compound distributions to be log-concave, and specific examples are presented illustrating all the above results.

Item Type: Article
Uncontrolled Keywords: cs.IT cs.IT math.IT math.PR 62B10, 94A17
Divisions: Div F > Signal Processing and Communications
Depositing User: Cron Job
Date Deposited: 08 Jan 2018 20:12
Last Modified: 18 Aug 2020 12:42