Abstract
The goals of this chapter are to: Explore the conditions under which there is equality between the Kantorovich and the Kantorovich–Rubinstein functionals; Provide inequalities between the Kantorovich and Kantorovich–Rubinstein functionals; Provide criteria for convergence, compactness, and completeness of probability measures in probability spaces involving the Kantorovich and Kantorovich–Rubinstein functionals; Analyze the problem of uniformity between the two functionals.
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Notes
- 1.
See Examples 3.3.3 and 4.3.2 in Chaps. 3 and 4, respectively.
- 2.
- 3.
See Example 3.3.2 in Chap. 3.
References
Billingsley P (1999) Convergence of probability measures, 2nd edn. Wiley, New York
Dudley RM (2002) Real analysis and probability, 2nd edn. Cambridge University Press, New York
Hennequin PL, Tortrat A (1965) Théorie des probabilités et quelques applications. Masson, Paris
Levin VL (1975) On the problem of mass transfer. Sov Math Dokl 16:1349–1353
Neveu J, Dudley RM (1980) On Kantorovich–Rubinstein theorems (transcript)
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Quantitative Relationships Between Minimal Distances and Minimal Norms. In: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4869-3_6
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DOI: https://doi.org/10.1007/978-1-4614-4869-3_6
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