Discrete versions of the transport equation and the Shepp-Olkin conjecture

Abstract

We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coefficients which allow us to characterize transport problems in a gradient flow setting, and form the basis of our introduction of a discrete version of the Benamou-Brenier formula. Further, we use these coefficients to state a new form of weighted log-concavity. These results are applied to prove the monotone case of the Shepp-Olkin entropy concavity conjecture.

Published at http://dx.doi.org/10.1214/14-AOP973 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Ricci curvature of finite Markov chains via convexity of the entropy
• Concavity of entropy under thinning
• A natural derivative on [0,n] and a binomial Poincaré inequality

Cited by in corpus (7)

• A discrete log-Sobolev inequality under a Bakry-Emery type condition
• A proof of the Shepp-Olkin entropy concavity conjecture
• A proof of the Shepp-Olkin entropy monotonicity conjecture
• A de Bruijn identity for symmetric stable laws
• Entropy and thinning of discrete random variables
• On the Maximum Entropy of a Sum of Independent Discrete Random Variables
• Maximum Entropy of Sums of Independent Ternary Random Variables