Large deviations and variational theorems for marginal problems
type: Item
instance of: scholarly article
title: Large deviations and variational theorems for marginal problems
zbMATH Open document ID: 0922.60029
DOI: 10.2307/3318614
author: Patrick Cattiaux
author: Fabrice Gamboa
published in: Bernoulli
publication date: 1999-10-03T00:00:00Z
review text: 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 \(P\).
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1275547
zbMATH Keywords: large deviations
zbMATH Keywords: marginal problems
reviewed by: Nina Gantert
MaRDI profile type: Publication
OpenAlex ID: W2066653135
description: scientific article; zbMATH DE number 1275547