The Annals of Probability

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

Erwan Hillion and Oliver Johnson

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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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 276-306.

Received: March 2013
Revised: September 2014
First available in Project Euclid: 2 February 2016

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 94A17: Measures of information, entropy 60D99: None of the above, but in this section

Entropy transportation of measures Bernoulli sums concavity


Hillion, Erwan; Johnson, Oliver. Discrete versions of the transport equation and the Shepp–Olkin conjecture. Ann. Probab. 44 (2016), no. 1, 276--306. doi:10.1214/14-AOP973.

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