Generalized Hoeffding-Fréchet functionals and mass transportation

Abstract This note is concerned with some historical remarks on and a partial review of two interesting mathematical subjects, the generalized Hoeffding-Fréchet functionals and the Monge-Kantorovich mass transportation problem. Both topics have a different motivation and history and are often considered in the literature as different subjects. The main aim of this review is to point out the close connection of these topics and to indicate some possibly fruitful relationships. For the class of Hoeffding-Fréchet functionals risk bounds for a lot of well-motivated additional model constraints have been worked out in recent years. These kinds of constraints are motivated by risk applications. We indicate some interesting connections of these developments to mass transportation problems as, e.g., to the solution of nonlinear mass transportation problems or to the use of stochastic ordering methods to the solution of mass transportation problems. We also briefly indicate the development of algorithms in the transportation problem by the regularization method (entropic optimal transport) and give corresponding references in the literature.