Distribution functions, extremal limits and optimal transport

Abstract

Encouraged by the study of extremal limits for sums of the formwith uniformly distributed sequences {xn}, {yn} the following extremal problem is of interestfor 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.

Bounds on integrals with respect to multivariate copulas

“…Just as Iacó at al. did in [10], we remark that this result can be used when the distributions µ 1 , . .…”

“…Our main result is a version of Theorems 2.1 and 2.2 from [9] for arbitrary dimensions along with a similar convergence result as Theorem 4.2 from [10]. To this end we start by introducing the concept of a multidimensional assignment problem.…”

“…The proof that the sequence of optimizers converges to an optimizer for the continuous function follows largely the arguments of the proof of Theorem 4.2 in [10]. To begin with, we can use Theorem 5.21 from [12] to deduce that C max n converges weakly, at least along some subsequence, to a copula C * .…”

“…The key to generalizing the method from [10] to higher dimensions now lies in recognizing the structural analogy between copulas and assignment problems.…”

“…Furthermore, Iacó et al [10] showed that the sequence of maximizers C max n converges, at least along some subsequence, to a maximizer C * of the problem…”