Distribution functions, extremal limits and optimal transport

Abstract

Encouraged by the study of extremal limits for sums of the form $$\lim_{N\to\infty}\frac{1 }{N}\sum_{n=1}^N c(x_n,y_n)$$ with uniformly distributed sequences $\{x_n\},\,\{y_n\}$ the following extremal problem is of interest $$\max_{\gamma}\int_{[0,1]^2}c(x,y)\gamma(dx,dy),$$ for probability measures $\gamma$ on the unit square with uniform marginals, i.e., measures whose distribution function is a copula. The aim of this article is to relate this problem to combinatorial optimization and to the theory of optimal transport. Using different characterizations of maximizing $\gamma$'s one can give alternative proofs of some results from the field of uniform distribution theory and beyond that treat additional questions. Finally, some applications to mathematical finance are addressed.