Distribution functions, extremal limits and optimal transport

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 $\max_γ\int_{[0,1]^2}c(x,y)γ(dx,dy),$ for probability measures $γ$ 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 $γ$'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.

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