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Mass transportation problems with capacity constraints

Part of: Distribution theory - Probability Multivariate analysis Distribution theory

Published online by Cambridge University Press:  14 July 2016

S. T. Rachev  and
I. Olkin
S. T. Rachev*
Affiliation:
University of California, Santa Barbara
I. Olkin*
Affiliation:
Stanford University
*
∗Postal address: Statistics and Applied Probability, University of California, Santa Barbara, CA 93106–3110, USA.
∗∗Postal address: Department of Statistics, Stanford University, Stanford, CA 94305–4065, USA. Supported in part by National Science Foundation.
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Abstract

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g. Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.

Keywords

Monge–Kantorovich problemmultivariate distributionsmarginal distributionsmeasures of dependencelinear programminggreedy algorithms

MSC classification

Primary: 60E05: Distributions 62E10: Characterization and structure theory
Secondary: 62H05: Characterization and structure theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 433 - 445
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Supported in part by Econometric Institute, Erasmus University, Rotterdam and the Alexander von Humboldt Foundation.

References

Appell, P. (1884). Mémoires sur les deblais et remblais des systèmes continués et discontinués. Mémoires des sav. etr., Ser. 2, 29, 3 mémoire.Google Scholar
Balinski, M. L., and Rachev, S. T. (1989). On Monge–Kantorovich problems. Preprint, SUNY, Stony Brook, Dept. of Applied Mathematics & Statistics. NSF-Grant Proposal, DMS-89 02330.Google Scholar
Barnes, E. R., and Hoffman, A. J. (1985). On transportation problems with upper bounds on leading rectangles. SIAM J. Alg. Discrete Methods 6, 487–496.Google Scholar
Cambanis, S., Simons, G., and Stout, W. (1976). Inequalities for Ek(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitsth. 36, 285–294.CrossRefGoogle Scholar
Cuesta-Albertos, J. A., Matran, C., Rachev, S. T. and Rüschendorf, L. (1996). Mass transportation problems in probability theory. Math. Scientist 21, 37–72.Google Scholar
Dall'Aglio, G. (1956). Sugli estremi dei momenti delle funzioni di ripartizione dopia. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, 33–74.Google Scholar
Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563–1572.Google Scholar
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks-Cole, Pacific Grove, CA.Google Scholar
Gini, C. (1914). Di una misura della dissomiglianza tra due gruppi di quantita e della sue applicazioni allo studio delle relazioni statistiche. Atti del Reale Instituo Veneto di Sci., Lettera et Arti. LXXIV (1914–1915), 185–213.Google Scholar
Hoffman, A. J. (1963). On simple linear programming problems. In Convexity ed. Klee, V. (Proc. Symp. Pure Math. 7). AMS, Providence, RI, pp. 317–327.Google Scholar
Hoffman, A. J., and Veinott, A. F. (1990). Staircase transportation problems with superadditive rewards and cumulative capacities. Preprint, Stanford University, Stanford, CA.Google Scholar
Hoffman, A. J., and Veinott, A. F. (1993). Staircase transportation problems with superadditive rewards and cumulative capacities. Math. Programming 62, 199–213.Google Scholar
Kakosyan, A. B., Klebanov, L., and Rachev, S. T. (1988). Quantitative Criteria for Convergence of Probability Measures. Ayastan, Erevan. (In Russian.)Google Scholar
Kalashnikov, V. V., and Rachev, S. T. (1990). Mathematical Methods for Construction of Stochastic Queueing Models. Wadsworth & Brooks-Cole, Pacific Grove, CA.Google Scholar
Kantorovich, L. V. (1942). On the transfer of masses. Dokl. Acad. Nauk. USSR 37, 227–229.Google Scholar
Kantorovich, L. V. (1948). On a problem of Monge. Usp. Mat. Nauk 3, 225–226. (In Russian.)Google Scholar
Kantorovich, L. V., and Akilov, G. P. (1984). Functional Analysis, Nauka, Moscow. (In Russian.)Google Scholar
Kruskal, W. H. (1958). Ordinal measures of association. J. Amer. Statist. Assoc. 53, 814–861.Google Scholar
Levin, V. L., and Rachev, S. T. (1989). New duality theorems for marginal problems with some applications in stochastics. In Lecture Notes in Math. 1412. Springer, New York, pp. 137–170.Google Scholar
Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais. In Histoire de l'Academie Royale des Sciences avec les memoires de mathematique et physique pour la mème année, pp. 666–704.Google Scholar
Olkin, I., and Rachev, S. T. (1990). Distributions with given marginals. Tech. Report 270, Dept. of Statistics, Stanford University, Stanford, CA.Google Scholar
Rachev, S. T. (1981). On minimal metrics in the space of real-valued random variables. Sov. Math. Dokl. 23, 425–432.Google Scholar
Rachev, S. T. (1984). Hausdorff metric construction in the probability measures space. Pliska, Studia Mathematica 7, 152–162.Google Scholar
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770–799.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems; Vol. 1, Theory; Vol. 2, Applications. Springer, New York (to appear).Google Scholar
Stoyan, P. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester.Google Scholar
Sudakov, V. N. (1976). Geometric problems in the theory of infinite-dimensional probability distributions. Tr. Mat. Inst. V. A. Steklov Akad. Nauk. SSSR 141. (In Russian.) (English translation (1979). Proc. Steklov Inst. Math. 2.)Google Scholar
Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.Google Scholar
Topkis, D. M., and Veinott, A. F. Jr. (1973). Monotone solution of extremal problems on lattices (Abstract). In Abstracts of 8th International Symposium on Mathematical Programming. Stanford University, Stanford, CA, p. 131.Google Scholar
Veinott, A. F. Jr. (1989). Representation of general and polyhedral sublattices and sublattices of product spaces. Linear Alg. Appl. 114/115, 681–704.Google Scholar