Large deviations and variational theorems for marginal problems

Abstract

On a product probability space (E ×F, P), we give variational characterizations for the existence of a probability measure Q with given marginals, such that Q is absolutely continuous with respect to P and its density satisfies some integrability conditions. These characterizations, which are in some sense the dual formulation of a theorem due to Strassen, are obtained by using large-deviations methods. We also study the minimal realizations of such Q.