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The Methods of Distances in the Theory of Probability and Statistics pp 11–31Cite as

Probability Distances and Probability Metrics: Definitions

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Abstract

The goals of this chapter are to: Provide examples of metrics in probability theory; Introduce formally the notions of a probability metric and a probability distance;

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Notes

  1. 1.

    Mostafaei and Kordnourie [2011] is a more recent general publication on probability metrics and their applications.

  2. 2.

    See Hennequin and Tortrat [1965].

  3. 3.

    The proof of this representation is given by [Dudley, 2002, p. 333] for the case p = 1.

  4. 4.

    See Lukacs [1968, Chap. 3] and Dudley [1976, Theorem 3.5].

  5. 5.

    See Massey [1950] and Thompson [1979].

  6. 6.

    See Thompson [1966].

  7. 7.

    See Hausdorff [1949].

  8. 8.

    A more detailed analysis of the metric H will be given in Sect. 4.2.

  9. 9.

    See Dunford and Schwartz [1988, Theorem 1.6.19].

  10. 10.

    Birnbaum and Orliz [1931] and Dunford and Schwartz [1988, p. 400]

  11. 11.

    If we replace “semidistance” with “distance,” then the statement continues to hold.

  12. 12.

    See [Dudley, 2002, Sect. 11.5].

  13. 13.

    See Billingsley [1968, Appendix III, p. 234]

  14. 14.

    See Cohn [1980, Corollary 8.2.17] and Dudley [2002, Proposition 13.2.5].

  15. 15.

    See Kaufman [1984].

  16. 16.

    See Dudley [2002, p. 347].

  17. 17.

    See Theorem 3.3.1 in Sect. 3.3.

  18. 18.

    See Loeve [1963, p. 99] and Dudley [2002, p. 82].

  19. 19.

    See Berkes and Phillip [1979].

  20. 20.

    See, for example, Loeve [1963, p. 99].

  21. 21.

    See Hewitt and Stromberg [1965, Theorems 22.7 and 22.8, p. 432–133].

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics College of Business, Stony Brook University, Stony Brook, NY, USA

    Svetlozar T. Rachev

  2. FinAnalytica, New York, NY, USA

    Svetlozar T. Rachev

  3. Department of Probability and Statistics, Charles University, Prague, Czech Republic

    Lev B. Klebanov

  4. EDHEC Business School EDHEC-Risk Institute, Singapore, Singapore

    Stoyan V. Stoyanov

  5. EDHEC Business School EDHEC-Risk Institute, Nice, France

    Frank J. Fabozzi

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  1. Svetlozar T. Rachev
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  2. Lev B. Klebanov
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  3. Stoyan V. Stoyanov
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  4. Frank J. Fabozzi
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Probability Distances and Probability Metrics: Definitions. 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_2

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