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Optimal couplings are totally positive and more

Part of: Basic linear algebra Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Paul Glasserman  and
David D. Yao
Paul Glasserman
Affiliation:
Graduate School of Business, Columbia University, New York, NY 10027, USA. Email address: pg20@columbia.edu
David D. Yao
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 10027-6699, USA. Email address: yao@columbia.edu
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Abstract

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

Keywords

Optimal couplingtotal positivityMonge sequence

MSC classification

Primary: 60B99: None of the above, but in this section
Secondary: 15A99: Miscellaneous topics
Type
Part 6. Stochastic processes
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 321 - 332
Copyright
Copyright © Applied Probability Trust 2004 

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