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How Close Are the Individual and Collective Models in Risk Theory?

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The Methods of Distances in the Theory of Probability and Statistics

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Abstract

The subject of this chapter is individual and collective models in insurance risk theory and how ideal probability metrics can be employed to calculate the distance between them.

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Notes

  1. 1.

    See Gerber [1981, p. 97].

  2. 2.

    See Gerber [1981, Chap. 4].

  3. 3.

    See Gerber [1981, Sect. 1, Chap. 4].

  4. 4.

    See Gerber [1981, p. 50].

  5. 5.

    See Gerber [1981, p. 97] for s = 1.

  6. 6.

    See, for example, Dunford and Schwartz [1988, Sect. IV.8] and Neveu [1965].

  7. 7.

    See condition (17.3.10) for \(\boldsymbol \zeta _{m,p}\) in Chap. 17.

  8. 8.

    See Definition 15.3.1 in Chap. 15.

  9. 9.

    See (15.3.1) in Chap. 15.

  10. 10.

    See, for example, (6.5.11) in Chap. 6.

  11. 11.

    See Lemmas 16.3.1 and 16.3.3 and (16.3.7).

  12. 12.

    See (18.2.13).

  13. 13.

    See Zolotarev [1986, Theorem 1.4.5] and Kalashnikov and Rachev [1988, Chap. 3, p. 10, Theorem 10].

  14. 14.

    See (18.2.3), (18.2.4), and (18.2.5).

  15. 15.

    See Theorem 18.2.1 and Remark 18.2.2.

  16. 16.

    See (18.2.4), (18.2.7), and (18.2.26).

References

  1. Arak TV, Zaitsev AYu (1988) Uniform limit theorems for sums of independent random variables. In: Proceedings of the Steklov Institute of Mathematics, vol 174, AMS

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  2. Dunford N, Schwartz J (1988) Linear operators, vol 1. Wiley, New York

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  3. Gerber H (1981) An introduction to mathematical risk theory. Huebner Foundation Monograph, Philadelphia

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  4. Kalashnikov VV, Rachev ST (1988) Mathematical methods for construction of stochastic queueing models. Nauka, Moscow (in Russian) [English transl., (1990) Wadsworth, Brooks–Cole, Pacific Grove, CA]

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  5. Neveu J (1965) Mathematical foundations of the calculus of probability. Holden-Day, San Francisco

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  6. Zolotarev VM (1986) Contemporary theory of summation of independent random variables. Nauka, Moscow (in Russian)

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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

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). How Close Are the Individual and Collective Models in Risk Theory?. 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_18

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