CMStatistics 2018: Start Registration
View Submission - CMStatistics
Title: A note on duality theorems in mass transportation Authors:  Pietro Rigo - University of Pavia (Italy) [presenting]
Abstract: In the framework of mass transportation, let $(\mathcal{X},\mathcal{F},\mu)$ and $(\mathcal{Y},\mathcal{G},\nu)$ be probability spaces and $c:\mathcal{X}\times\mathcal{Y}\rightarrow\mathbb{R}$ a measurable cost function such that $f_1+g_1\le c\le f_2+g_2$ for some $f_1,\,f_2\in L_1(\mu)$ and $g_1,\,g_2\in L_1(\nu)$. Define $\alpha(c)=\inf_P\int c\,dP$ and $\alpha^*(c)=\sup_P\int c\,dP$, where $\inf$ and $\sup$ are over the probabilities $P$ on $\mathcal{F}\otimes\mathcal{G}$ with marginals $\mu$ and $\nu$. A few duality theorems for $\alpha(c)$ and $\alpha^*(c)$, not requiring $\mu$ or $\nu$ to be perfect, are proved. As an example, suppose $\mathcal{X}$ and $\mathcal{Y}$ are metric spaces and at least one of $\mu$ and $\nu$ is separable. Then, duality holds for $\alpha(c)$ (for $\alpha^*(c)$) provided $c$ is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both $\alpha(c)$ and $\alpha^*(c)$ if the sections $x\mapsto c(x,y)$ and $y\mapsto c(x,y)$ are continuous. This improves the existing results if $c$ has continuous sections and the cardinalities of $\mathcal{X}$ and $\mathcal{Y}$ do not exceed the continuum. Finally, the duality problem is investigated in a finitely additive setting. In this case, if $c$ is bounded, some results by Ruschendorf are generalized.