Characterization of a class of weak transport-entropy inequalities on the line

Abstract

We study an optimal weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.Date: August 3, 2018. 1991 Mathematics Subject Classification. 60E15, 32F32 and 26D10.

On a mixture of Brenier and Strassen Theorems

“…In particular, Kantorovich type duality formulas are obtained [33,Theorem 9.6] under the assumption that c is convex with respect to the p variable (and some additional mild regularity conditions). We refer to [22,31,32,66,68] for works directly connected to [33] and to [65] for an up-to-date survey of applications of weak transport costs to concentration of measure. Besides their many applications in the field of functional inequalities and concentration of measure, it turns out that weak transport costs are also interesting in themselves as a natural generalization of the transportation problem.…”

“…In particular, Kantorovich type duality formulas are obtained [, Theorem 9.6] under the assumption that is convex with respect to the variable (and some additional mild regularity conditions). We refer to for works directly connected to and to for an up‐to‐date survey of applications of weak transport costs to concentration of measure.…”

Rationalizing Rational Expectations? Tests and Deviations

“…By Strassen's theorem (Strassen, 1965, Theorem 8), the existence of (Y, ψ) with margins equal to F Y and F ψ and such that E [Y |ψ] = ψ is equivalent to f dF ψ ≤ f dF Y for every convex function f . By, e.g., Proposition 2.3 in Gozlan et al (2018), this is, in turn, equivalent to (iii).…”

Weak monotone rearrangement on the line

“…[14,Proposition 2.6] we know that ∀k ≤ n : ≤k x 2 ≤ ≤k x 1 , ≤k z 1 ≤ ≤k z 2 . But then also ≤k x 2 − z 2 ≤ ≤k x 1 − z 1 , so again by [14,Proposition 2.6] we conclude (id − T 1 )(η 1 ) ≤ c (id − T 2 )(η 2 ). The second statement easily follows from the first one.…”