Open Access
May 2010 Relative log-concavity and a pair of triangle inequalities
Yaming Yu
Bernoulli 16(2): 459-470 (May 2010). DOI: 10.3150/09-BEJ216


The relative log-concavity ordering ≤lc between probability mass functions (pmf’s) on non-negative integers is studied. Given three pmf’s f, g, h that satisfy f ≤lcg ≤lch, we present a pair of (reverse) triangle inequalities: if ∑iifi=∑iigi<∞, then


and if ∑iigi=∑iihi<∞, then


where D(⋅|⋅) denotes the Kullback–Leibler divergence. These inequalities, interesting in themselves, are also applied to several problems, including maximum entropy characterizations of Poisson and binomial distributions and the best binomial approximation in relative entropy. We also present parallel results for continuous distributions and discuss the behavior of ≤lc under convolution.


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Yaming Yu. "Relative log-concavity and a pair of triangle inequalities." Bernoulli 16 (2) 459 - 470, May 2010.


Published: May 2010
First available in Project Euclid: 25 May 2010

zbMATH: 1248.60028
MathSciNet: MR2668910
Digital Object Identifier: 10.3150/09-BEJ216

Keywords: Bernoulli sum , binomial approximation , Hoeffding’s inequality , maximum entropy , minimum entropy , negative binomial approximation , Poisson approximation , Relative entropy

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability

Vol.16 • No. 2 • May 2010
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