Relative log-concavity and a pair of triangle inequalities

TL;DR

This paper investigates the relative log-concavity ordering among probability mass functions on non-negative integers, establishing new triangle inequalities involving Kullback-Leibler divergence and applying these to characterize distributions and approximation problems.

Contribution

It introduces a pair of reverse triangle inequalities for the relative log-concavity order and applies them to distribution characterization and approximation problems.

Findings

Established reverse triangle inequalities involving KL divergence.

Applied inequalities to maximum entropy characterizations of Poisson and binomial distributions.

Extended results to continuous distributions and convolution behaviors.

Abstract

The relative log-concavity ordering $\leq_{\mathrm{lc}}$ between probability mass functions (pmf's) on non-negative integers is studied. Given three pmf's $f,g,h$ that satisfy $f\leq_{\mathrm{lc}}g\leq_{\mathrm{lc}}h$, we present a pair of (reverse) triangle inequalities: if $\sum_iif_i=\sum_iig_i<\infty,$ then \[D(f|h)\geq D(f|g)+D(g|h)\] and if $\sum_iig_i=\sum_iih_i<\infty,$ then \[D(h|f)\geq D(h|g)+D(g|f),\] where $D(\cdot|\cdot)$ 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 $\leq_{\mathrm{lc}}$ under convolution.