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Series expansions for convolutions of Pareto distributions

Author

Listed:
  • Nguyen Quang Huy

    (Institut de Science Financière et d'Assurances, Université Lyon 1, 50 avenue Tony Garnier, 69366 Lyon cedex 7, France)

  • Robert Christian Y.

    (Institut de Science Financière et d'Assurances, Université Lyon 1, 50 avenue Tony Garnier, 69366 Lyon cedex 7, France)

Abstract

Asymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.

Suggested Citation

  • Nguyen Quang Huy & Robert Christian Y., 2015. "Series expansions for convolutions of Pareto distributions," Statistics & Risk Modeling, De Gruyter, vol. 32(1), pages 49-72, April.
    • Handle: RePEc:bpj:strimo:v:32:y:2015:i:1:p:49-72:n:3
    DOI: 10.1515/strm-2014-1168
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    References listed on IDEAS

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    1. Albrecher, Hansjörg & Kortschak, Dominik, 2009. "On ruin probability and aggregate claim representations for Pareto claim size distributions," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 362-373, December.
    2. Kortschak, Dominik & Albrecher, Hansjörg, 2010. "An asymptotic expansion for the tail of compound sums of Burr distributed random variables," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 612-620, April.
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