Entropy-variance inequalities for discrete log-concave random variables via degree of freedom

Abstract

We utilize a discrete version of the notion of degree of freedom to prove a sharp min-entropy-variance inequality for integer valued log-concave random variables. More specifically, we show that the geometric distribution minimizes the min-entropy within the class of log-concave probability sequences with fixed variance. As an application, we obtain a discrete Rényi entropy power inequality in the log-concave case, which improves a result of Bobkov, Marsiglietti and Melbourne (2022).

arXiv commentThe final version uploaded; To appear in Discrete Mathematics

References in corpus (9)

• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Log-concavity and the maximum entropy property of the Poisson distribution
• Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
• A lower bound on the differential entropy of log-concave random vectors with applications
• Preservation of log-concavity on summation
• Asymptotics of the discrete log-concave maximum likelihood estimator and related applications
• Majorization and Rényi Entropy Inequalities via Sperner Theory
• Sharp moment-entropy inequalities and capacity bounds for log-concave distributions
• Bernoulli sums and Rényi entropy inequalities