Large deviations and variational theorems for marginal problems
Given a product probability space \((E \times F, P)\) and two probability measures \(\mu\) and \(\nu\) defined on \(E\) and \(F\), respectively, the authors investigate whether there exists a probability measure \(Q\) on \(E\times F\) with marginals \(\mu\) and \(\nu\) such that \(Q\) is absolutely continuous with respect to \(P\), and the Radon-Nikodym density \(dQ/dP\) satisfies some integrability conditions. They give necessary and sufficient conditions in terms of a variational characterization of \(P\). For a weakly closed, convex subset \(\Lambda\) of the probability distributions on \(E\times F\), Strassen's theorem gives a necessary and sufficient condition for the existence of a \(Q \in \Lambda\) with marginals \(\mu\) and \(\nu\). The variational characterization obtained by the authors can be read as a dual formulation of Strassen's condition. The proof is based on large deviation methods. The authors also study the minimal realizations of such a \(Q\). A well-known example is entropy-minimization with given marginals, where one looks for \(Q\) with marginals \(\mu\) and \(\nu\) which minimizes the relative entropy with respect to \(P\).
scientific article; zbMATH DE number 1275547