On the n-coupling problem

Abstract

In this paper we obtain based on an idea of M. Knott and C.S. Smith (1994, Linear Algebra Appl. 199, 363-371) characterizations of solutions of three-coupling problems by reduction to the construction of optimal couplings of each of the variables to the sum. In the case of normal distributions this leads to a complete solution. Under a technical condition this idea also works for general distributions and one obtains explicit results. We extend these results to the n-coupling problem and derive a characterization of optimal n-couplings by several 2-coupling problems. This leads to some constructive existence results for Monge solutions. © 2002 Elsevier Science (USA).