The Annals of Probability

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

Erwan Hillion and Oliver Johnson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • [1] Amari, S.-i. and Nagaoka, H. (2000). Methods of Information Geometry. Translations of Mathematical Monographs 191. Amer. Math. Soc., Providence, RI.
  • [2] Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. Birkhäuser, Basel.
  • [3] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses 10. Société Mathématique de France, Paris.
  • [4] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/84. Lecture Notes in Math. 1123 177–206. Springer, Berlin.
  • [5] Benamou, J.-D. and Brenier, Y. (1999). A numerical method for the optimal time-continuous mass transport problem and related problems. In Monge Ampère Equation: Applications to Geometry and Optimization (Deerfield Beach, FL, 1997). Contemp. Math. 226 1–11. Amer. Math. Soc., Providence, RI.
  • [6] Benamou, J.-D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84 375–393.
  • [7] Bickel, P. J. and Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196–1217.
  • [8] Carlen, E. A. and Gangbo, W. (2003). Constrained steepest descent in the 2-Wasserstein metric. Ann. of Math. (2) 157 807–846.
  • [9] Cordero-Erausquin, D. (2002). Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161 257–269.
  • [10] Cover, T. M. and Thomas, J. A. (1991). Elements of Information Theory. Wiley, New York.
  • [11] Efron, B. (1965). Increasing properties of Pólya frequency functions. Ann. Math. Statist. 36 272–279.
  • [12] Erbar, M. and Maas, J. (2012). Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206 997–1038.
  • [13] Gozlan, N., Roberto, C., Samson, P.-M. and Tetali, P. (2014). Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Related Fields 160 47–94.
  • [14] Hillion, E. (2012). Concavity of entropy along binomial convolutions. Electron. Commun. Probab. 17 9.
  • [15] Hillion, E. (2014). Contraction of Measures on Graphs. Potential Anal. 41 679–698.
  • [16] Hillion, E., Johnson, O. T. and Yu, Y. (2014). A natural derivative on $[0,n]$ and a binomial Poincaré inequality. ESAIM Probab. Statist. 18 703–712.
  • [17] Johnson, O. (2007). Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 791–802.
  • [18] Johnson, O. T. and Suhov, Y. M. (2003). The von Neumann entropy and information rate for integrable quantum Gibbs ensembles 2. Quantum Computers and Computing 4 128–143.
  • [19] Jordan, R., Kinderlehrer, D. and Otto, F. (1998). The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29 1–17.
  • [20] Kontoyiannis, I., Harremoës, P. and Johnson, O. (2005). Entropy and the law of small numbers. IEEE Trans. Inform. Theory 51 466–472.
  • [21] Liggett, T. M. (1997). Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A 79 315–325.
  • [22] Lott, J. and Villani, C. (2009). Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 903–991.
  • [23] Maas, J. (2011). Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 2250–2292.
  • [24] Mateev, P. (1978). The entropy of the multinomial distribution. Teor. Verojatnost. i Primenen. 23 196–198.
  • [25] Mielke, A. (2013). Geodesic convexity of the relative entropy in reversible Markov chains. Calc. Var. Partial Differential Equations 48 1–31.
  • [26] Niculescu, C. P. (2000). A new look at Newton’s inequalities. JIPAM. J. Inequal. Pure Appl. Math. 1 Article 170.
  • [27] Pemantle, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41 1371–1390.
  • [28] Rényi, A. (1957). A characterization of Poisson processes. Magyar Tud. Akad. Mat. Kutató Int. Közl. 1 519–527.
  • [29] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.
  • [30] Shepp, L. A. and Olkin, I. (1981). Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution. In Contributions to Probability 201–206. Academic Press, New York.
  • [31] Sturm, K.-T. (2006). On the geometry of metric measure spaces. I. Acta Math. 196 65–131.
  • [32] Sturm, K.-T. (2006). On the geometry of metric measure spaces. II. Acta Math. 196 133–177.
  • [33] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [34] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin.
  • [35] Yu, Y. (2008). On the maximum entropy properties of the binomial distribution. IEEE Trans. Inform. Theory 54 3351–3353.
  • [36] Yu, Y. (2009). Monotonic convergence in an information-theoretic law of small numbers. IEEE Trans. Inform. Theory 55 5412–5422.
  • [37] Yu, Y. and Johnson, O. T. (2009). Concavity of entropy under thinning. In Proceedings of ISIT 2009, 28th June–3rd July 2009, Seoul 144–148. IEEE, New York.