Duality and perfect probability spaces
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- by D. Ramachandran and L. Rüschendorf PDF
- Proc. Amer. Math. Soc. 124 (1996), 2223-2228 Request permission
Abstract:
Given probability spaces $(X_i,\mathcal {A}_i,P_i), i=1,2,$ let $\mathcal {M}(P_1,P_2)$ denote the set of all probabilities on the product space with marginals $P_1$ and $P_2$ and let $h$ be a measurable function on $(X_1 \times X_2,\mathcal {A}_1 \otimes \mathcal {A}_2).$ Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinštein (1958) for the case of compact metric spaces are concerned with the validity of the duality \begin{align*} &\sup \{ \int h dP: P \in \mathcal {M}(P_1,P_2) \} \ &\qquad = \: \inf \{ \sum _{i=1}^{2} \int h_i dP_i : h_i \in \mathcal {L}^1 (P_i) \; \; and \; \; h \leq {\oplus }_i h_i\} \end{align*} (where $\mathcal {M}(P_1,P_2)$ is the collection of all probability measures on $(X_1 \times X_2, \mathcal {A}_1 \otimes \mathcal {A}_2)$ with $P_1$ and $P_2$ as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.References
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Additional Information
- D. Ramachandran
- Affiliation: Department Of Mathematics and Statistics, California State University, 6000 J Street, Sacramento, California 95819-6051
- Email: chandra@csus.edu
- L. Rüschendorf
- Affiliation: California State University, Sacramento and Universität Freiburg
- Address at time of publication: Institut für Mathematische Stochastik, Albert-Ludwigs-Universität, Hebelstr. 27, D-79104 Freiburg, Germany
- Email: ruschen@buffon.mathematik.uni-freiburg.de
- Received by editor(s): December 15, 1994
- Additional Notes: Research supported in part by an Internal Awards Grant from the California State University, Sacramento
- Communicated by: Richard T. Durrett
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2223-2228
- MSC (1991): Primary 60A10, 28A35
- DOI: https://doi.org/10.1090/S0002-9939-96-03462-4
- MathSciNet review: 1342043