Developments on Fréchet-bounds
Ludger Rüschendorf · Lecture notes-monograph series · 1996
This paper describes some developments in the field of Frechet bounds since the last report given at the Rome conference in 1990.At first a review is given on product type representations of distributions with given marginals and their relations to the solution of Schrδdinger type equations.The iterative proportional fitting procedure allows an approximate construction.Its convergence has been shown recently.We give a convergence proof of a modified algorithm under alternative assumptions.In the next part of the paper several sufficient and necessary conditions are given for the explicit construction of optimal multivariate couplings or, equivalently, tranportation plans.These results allow to calculate several multivariate examples, in particular examples for minimal £ pmetrics.In the final part we consider some recent examples of optimal couplings under additional or relaxed restrictions.We discuss a problem involving order restrictions, the case of fixed difference of the marginals, the application of a duality principle for Monge functions and Frechet bounds for marginal classes majorized by a finite measure.