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Discrete versions of the transport equation and the Shepp-Olkin conjecture Hillion, Erwan ; in Annals of Probability (2016), 44(1), 276-306 We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coeffcients which allow us to characterise transport problems in a gradient now setting ... [more ▼] We introduce a framework to consider transport problems for integer-valued random variables. We introduce weighting coeffcients which allow us to characterise transport problems in a gradient now setting, and form the basis of our introduction of a discrete version of the Benamou--Brenier formula. Further, we use these coeffcients 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. [less ▲] Detailed reference viewed: 151 (9 UL)A natural derivative on [0, n] and a binomial Poincaré inequality Hillion, Erwan ; ; in ESAIM: Probability and Statistics (2014), 18 We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We ... [more ▼] We consider probability measures supported on a finite discrete interval [0, n]. We introduce a new finite difference operator ∇n, defined as a linear combination of left and right finite differences. We show that this operator ∇n plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport of probability measures. [less ▲] Detailed reference viewed: 113 (2 UL)W1,+-interpolation of probability measures on graphs Hillion, Erwan in Electronic Journal of Probability (2014), 19 We generalize an equation introduced by Benamou and Brenier and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the ... [more ▼] We generalize an equation introduced by Benamou and Brenier and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a nal distributions (f_0(x)), (f_1(x)), we prove the existence of a curve (f_t(x)) satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem. [less ▲] Detailed reference viewed: 127 (4 UL)Contraction of Measures on Graphs Hillion, Erwan in Potential Analysis (2014), 41 Given a finitely supported probability measure μ on a connected graph G, we construct a family of probability measures interpolating the Dirac measure at some given point o ∈ G and μ. Inspired by Sturm ... [more ▼] Given a finitely supported probability measure μ on a connected graph G, we construct a family of probability measures interpolating the Dirac measure at some given point o ∈ G and μ. Inspired by Sturm-Lott-Villani theory of Ricci curvature bounds on measured length spaces, we then study the convexity of the entropy functional along such interpolations. Explicit results are given in three canonical cases, when the graph G is either Z^n , a cube or a tree. [less ▲] Detailed reference viewed: 128 (3 UL)Concavity of entropy along binomial convolution Hillion, Erwan in Electronic communications in probability (2012), 17 Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity ... [more ▼] Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity in t of the entropy of the convolution of a probability measure a, which has the law of a sum of independent Bernoulli variables, by the binomial measure of parameters n and t. [less ▲] Detailed reference viewed: 70 (3 UL) |
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