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Duality and perfect probability spaces

Authors: D. Ramachandran and L. Rüschendorf
Journal: Proc. Amer. Math. Soc. 124 (1996), 2223-2228
MSC (1991): Primary 60A10, 28A35
MathSciNet review: 1342043
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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-Rubin\v{s}tein (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 [Enhancements On Off] (What's this?)

  • 1. R. M. Dudley, Probabilities and metrics, Matematisk Institut, Aarhus Universitet, Aarhus, 1976. Convergence of laws on metric spaces, with a view to statistical testing; Lecture Notes Series, No. 45. MR 0488202
  • 2. Gnedenko, B.V., and Kolomogorov, A.N. (1954). Limit distributions for sums of independent random variables. Addison Wesley, Cambridge. MR 16:52d
  • 3. L. V. Kantorovič and G. Š. Rubinšteĭn, On a space of completely additive functions, Vestnik Leningrad. Univ. 13 (1958), no. 7, 52–59 (Russian, with English summary). MR 0102006
  • 4. Hans G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 4, 399–432. MR 761565,
  • 5. Monge, G. (1781). Mémoire sur la théorie des déblais ét ramblais. Mem. Math. Phys. Acad. Roy. Sci. Paris, 666-704.
  • 6. Jacques Neveu, Mathematical foundations of the calculus of probability, Translated by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. MR 0198505
  • 7. Jan K. Pachl, Two classes of measures, Colloq. Math. 42 (1979), 331–340. MR 567571,
  • 8. Jan K. Pachl, Correction to: “Two classes of measures” [Colloq. Math. 42 (1979), 331–340; MR 82b:28012], Colloq. Math. 45 (1981), no. 2, 331–333. MR 665799
  • 9. Svetlozar T. Rachev, Probability metrics and the stability of stochastic models, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991. MR 1105086
  • 10. Ramachandran, D. (1979). Perfect Measures, I and II. ISI Lecture Notes Series, 5 and 7, New Delhi, Macmillan. MR 81h:6005b; MR 81h:6005a
  • 11. Ramachandran, D. and Rüschendorf, L. (1995). A general duality theorem for marginal problems. Probab. Theory Relat. Fields, 101, 311-319.
  • 12. V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423–439. MR 177430,

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Additional Information

D. Ramachandran
Affiliation: Department Of Mathematics and Statistics, California State University, 6000 J Street, Sacramento, California 95819-6051

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

Keywords: Duality theorem, marginals, perfect measure, Marczewski function
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
Article copyright: © Copyright 1996 American Mathematical Society