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Duality theorems for Kantorovich-Rubinstein and Wasserstein functionals

Seria
Rozprawy Matematyczne tom/nr w serii: 299 wydano: 1990
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Abstrakty
EN

CONTENTS
§0. Introduction...................................................................................................................................5
§1. Notation and terminology..............................................................................................................6
§2. A generalization of the Kantorovich-Rubinstein theorem..............................................................8
§3. Application: explicit representations for a class of probability metrics.........................................14
§4. Topology of the Kantorovich-Rubinstein norm............................................................................18
§5. Dual representation for the Wasserstein functional....................................................................21
§6. Comparison of Wasserstein functional and Kantorovich-Rubinstein norm; completeness..........27
§7. Convergence of empirical measures; results of Fortet-Mourier type..........................................30
§8. The convex set of optimal measures..........................................................................................32
References......................................................................................................................................34
Słowa kluczowe
Tematy
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Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 299
Liczba stron
35
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCXCIX
Daty
wydano
1990
Twórcy
autor
  • University of California, Department of Statistics and Applied Probability, Santa Barbara, California 93106, USA
autor
  • University of California, Department of Statistics and Applied Probability, Santa Barbara, California 93106, USA
Bibliografia
  • [1] R. Bartoszyński, A characterization of weak convergence of measures, Ann. Math. Stat. 32 (1961), 561-576.
  • [2] P. Billingsley, Convergence of Probability Measures, John Wiley, New York 1968.
  • [3] J. P. R. Christensen, Topology and Borel structure, North-Holland-Elsevier, Amsterdam 1974.
  • [4] D. L. Cohn, Measure Theory, Birkhäuser, Boston 1980.
  • [5] A. de Acosta, Invariance principles for triangle arrays of B-valued random vectors, Ann. Probab. 10 (1982), 346-373.
  • [6] R. M. Dudley, Probabilities and metrics, Aarhus Universitet, Aarhus 1976.
  • [7] X. Fernique, Sur le Théorème de Kantorovich-Rubinstein dans les espaces polonaises, Lecture Notes in Math. 850, Springer-Verlag, 1981, pp. 6-10.
  • [8] R. Fortet and E. Mourier, Convergence de la repartition empirique vers la repartition théorétique, Ann. Sci. École Norm. Sup. 70 (3) (1953), 267-285.
  • [9] C. R. Givens and R. M. Shortt, A class of Wasserstein metrics for probability distributions, Michigan Math. J. 31 (1984), 231-240.
  • [10] L. V. Kantorovich and G. S. Rubinstein, On a space of completely additive functions (in Russian), Vestnik Leningrad. Univ. 13 (7) (1958), 52-59.
  • [11] H. G. Kellerer, Duality theorems for marginal problems, Z. Wahrsch. Verw. Gebiete 67 (1984), 399-432.
  • [12] H. G. Kellerer, Duality theorems and probability metrics, in: Proc. 7th Brasov Conf. 1982, Bucuresti 1984, pp. 211-220.
  • [13] H. G. Kellerer, Measure-theoretic versions of linear programming, Math. Z. 198 (1988), 367-400.
  • [14] J. H. B. Kemperman, On the role of duality in the theory of moments, in: Proc. Semi-infinite Programming and Applications 1981, Lecture Notes in Economics and Math. "Systems 215, Springer-Verlag, New York 1983, pp. 63-92.
  • [15] V. L. Levin and A. A. Milyutin, The problem of mass transfer with a discontinuous cost function, Russian Math. Surveys 34 (3) (1979), 1-78.
  • [16] V. L. Levin, The problem of mass transfer in a topological space, Soviet Math. Dokl. (1984), 638-643.
  • [17] J. Neveu and R. M. Dudley, On Kantorovich-Rubinstein theorems, typescript.
  • [18] S. T. Rachev, The Monge-Kantorovich mass transfer problem and its stochastic applications, Theory Prob. Appl. 29 (1984), 647-676.
  • [19] S. T. Rachev, Extreme functionals in the space of probability measures, in: Proc. Stability problems for stochastic models 1984, Lecture Notes in Math. 1155, Springer-Verlag, New York 1985, pp. 320-348.
  • [20] R. Ranga Rao, Relations between weak and uniform convergence of measures with applications, Ann. Math. Stat. 33 (1962), 659-680.
  • [21] R. M. Shortt, Strassen's marginal problem in two or more dimensions, Z. Wahrsch. Verw. Gebiete 64 (1983), 313-325.
  • [22] V. Strassen, The existence of probability measures with given marginals, Ann. Math. Stat. 36 (1965), 423-439.
  • [23] A. Szulga, On the Wasserstein metric, in: Transactions 8th Prague Conf. on Information Theory, Akademia, Prague 1978, pp. 267-273.
  • [24] A. Szulga, On minimal metrics in the space of random variables, Theory Prob. Appl. 27 (1982), 424-430.
  • [25] S. S. Vallander, Calculation of the Wasserstein distance between probability distributions on the line, Theory Prob. Appl. 18 (1973), 784-786.
  • [26] R. G. Douglas, On extremal measures and subspace density, Michigan Math. J. 11 (1964), 243-246.
  • [27] J. Lindenstrauss, A remark on doubly-stochastic measures, Amer. Math. Monthly 72 (1965), 379-382.
  • [28] R. M. Shortt, The singularity of extremal measures, Real Analysis Exchange 12 (1986-7), 205-215.
Języki publikacji
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Identyfikator YADDA
bwmeta1.element.zamlynska-96c9f8aa-610a-48db-94a1-2c0f49ba2f19
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ISBN
83-01-09970-4
ISSN
0012-3862
Kolekcja
DML-PL
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