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Supports of Extremal Doubly Stochastic Measures

Published online by Cambridge University Press:  20 November 2018

Abbas Moameni
Abbas Moameni*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON KIS 5B6 e-mail: [email protected]
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Abstract

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A doubly stochastic measure on the unit square is a Borel probability measure whose horizontal and vertical marginals both coincide with the Lebesgue measure. The set of doubly stochasticmeasures is convex and compact so its extremal points are of particular interest. The problem number 111 of Birkhoò is to provide a necessary and suõcient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Beneš and Štepán that an extremal doubly stochastic measure is concentrated on a set which admits an aperiodic decomposition. Hestir and Williams later found a necessary condition which is nearly sufficient by further refining the aperiodic structure of the support of extremal doubly stochastic measures. Our objective in this work is to provide a more practical necessary and nearly sufficient condition for a set to support an extremal doubly stochastic measure.

Keywords

49Q15optimal mass transportdoubly stochastic measuresextremalityuniqueness
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 381 - 391
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Ahmad, N., Kim, H. K., and McCann, R. J., Optimal transportation, topology and uniqueness. Bull. Math. Sci. 1(2011), no. 1,13-32.Google Scholar
[2] Aliprantis, C. D. and Border, K. C., Infinite dimensional analysis. Third edition. Springer, Berlin, 2006. http://dx.doi.org/10.1007/s13373-011-0002-7 Google Scholar
[3] Benes, V. and J. Stëpân, The support of extremal probability measures with given marginals. In: Mathematical statistics and probability theory, Vol. A, Reidel, Dordrecht, 1987, pp. 33–41.Google Scholar
[4] Bianchini, S. and Caravenna, L., On the extremality, uniqueness and optimality of transference plans. Bull. Inst. Math. Acad. Sin. (N.S.) 4(2009), no. 4, 353–454.Google Scholar
[5] Bianchini, S. and Caravenna, L., On optimality of c- cyclically monotone transference plans. C. R. Math. Acad. Sci. Paris 348 (2010), no. 11-12, 613–618.Google Scholar
[6] Birkhoff, G., Très observacionessobre el algebra lineal. Univ. Nac.Tucumân. Revista A (1946) 147–151.Google Scholar
[7] Birkhoff, G., Lattice theory Revised edition. American Mathematical Society, New York, 1948.Google Scholar
[8] Bogachev, V. I., Measure theory. Vol.I, II. Springer-Verlag, Berlin, 2007.Google Scholar
[9] Chiappori, P.-A., McCann, R. J., and Nesheim, L. P., Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness. Econom. Theory 42(2010), no. 2, 317–354. http://dx.doi.org/10.1007/s00199-009-0455-z Google Scholar
[10] Denny, J. L., The support of discrete extremal measures with given marginals. Michigan Math. J. 27(1980), no. 1, 59–64.Google Scholar
[11] Douglas, R. D., Onextremal measures and subspace density. Michigan Math. J. 11(1964) 243–246. http://dx.doi.org/10.1307/mmjV1029002309 Google Scholar
[12] Durante, F., J. Fernandez Sanchez, and Trutschnig, W., Multivariate copulas with hairpin support. J. Multivariate Anal. 130(2014), 323–334. http://dx.doi.org/10.1016/j.jmva.2O14.06.009 Google Scholar
[13] Hestir, K. and Williams, S. C., Supports of doubly stochastic measures.Bernoulli 1(1995), no. 3, 217–243. http://dx.doi.org/10.2307/3318478 Google Scholar
[14] Kaminski, A., Mikusinski, P., Sherwood, H., and Taylor, M. D., Doubly stochastic measures, topology, and latticework hairpins. J. Math. Anal. Appl. 152(1990), no. 1, 252–268. http://dx.doi.org/!0.1016/0022-247X(90)90102-L Google Scholar
[15] Lindenstrauss, J., A remark on extreme doubly stochastic measures. Amer. Math. Monthly 72(1965) 379–382.Google Scholar
[16] Losert, V. Counterexamples to some conjectures about doubly stochastic measures. Pacific J. Math. 99(1982), no. 2, 387–397. http://dx.doi.org/10.2307/2313497 Google Scholar
[17] Moameni, A., A characterization for solutions of the Monge-Kantorovich mass transport problem. Math. Ann., to appear, http://dx.doi.org/10.1007/s00208-015-1312-y Google Scholar
[18] McCann, R., Moameni, A., and Rifford, L., Uniqueness results for Monge-Kantorovich mass transport problems. http://people.math.carleton.ca/∼momeni/Publications.html.Google Scholar
[19] Nelsen, R. B., An introduction to copulas.Second edition. Springer Series in Statistics, Springer, New York, 2006.Google Scholar
[20] von Neumann, J., A certain zero-sum two-person game equivalent to the optimal assignment problem. In: Contributions to the theory of games, Vol. 2, Princeton, 1953, pp. 5–12.Google Scholar
[21] Quesada Molina, J. J. and J.-A. Rodriguez Lallena, Some remarks on the existence of doubly stochastic measures with latticework hairpin support. Aequationes Math. 47(1994), no. 2-3, 164–174. http://dx.doi.org/10.1007/BF01832957 Google Scholar
[22] Rachev, S. T. and Riischendorf, L., Mass transportation problems. Vol. I. Theory. Probability and its Applications (New York). Springer-Verlag, New York, 1998.Google Scholar
[23] Seethoffand, T. L. Shiflett, R. C., Doubly stochastic measures with prescribed support. Z. Wahrscheinlichkeitstheorie und Verw.Gebiete 41(1977/78), no. 4, 283–288. http://dx.doi.org/10.1007/BF00533599 Google Scholar
[24] Sherwood, H. and Taylor, M. D., Doubly stochastic measures with hairpin support.Probab. Theory Related Fields 78(1988), no. 4, 617–626. http://dx.doi.org/1 0.1007/BF00353879 Google Scholar
[25] Villani, C., Optimal transport, old and new.Grundlehren der MathematischenWissenschaften, 338. Springer-Verlag, Berlin, 2009.Google Scholar