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Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle

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Abstract

This chapter begins with an application of the theory of probability metrics to the problem of convergence of the empirical probability measure.

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Notes

  1. 1.

    See Prokhorov [1956].

  2. 2.

    See, for example, Billingsley [1999, Theorem 10.1].

  3. 3.

    See, for example, Billingsley [1999].

  4. 4.

    Given (i), then (ii) is equivalent to (12.3.5); see Kruglov [1973, Theorem 1].

  5. 5.

    See Bernstein [1964, p. 358].

  6. 6.

    See Bernstein [1964], Kruglov [1973], and de Acosta and Gine [1979].

  7. 7.

    See Gikhman and Skorokhod [1971, p. 491, or p. 416 of the English edition].

  8. 8.

    See Billingsley [1999].

  9. 9.

    See Billingsley [1999, Chap. 3].

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Author information

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

Authors
  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). Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle. 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_12

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