IDEAS home Printed from https://ideas.repec.org/p/arx/papers/math-0703022.html
   My bibliography  Save this paper

Tails of random sums of a heavy-tailed number of light-tailed terms

Author

Listed:
  • Christian Y. Robert
  • Johan Segers

Abstract

The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions allow the claim number distribution to be of consistent variation or to be in the domain of attraction of a Gumbel distribution with a mean excess function that grows to infinity sufficiently fast. Moreover, the claim number is not necessarily required to be independent of the claim sizes.

Suggested Citation

  • Christian Y. Robert & Johan Segers, 2007. "Tails of random sums of a heavy-tailed number of light-tailed terms," Papers math/0703022, arXiv.org, revised Oct 2007.
    • Handle: RePEc:arx:papers:math/0703022
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/math/0703022
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Cline, D. B. H. & Samorodnitsky, G., 1994. "Subexponentiality of the product of independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 49(1), pages 75-98, January.
    2. Sidney Resnick & Gennady Samorodnitsky, 1997. "Performance Decay in a Single Server Exponential Queueing Model with Long Range Dependence," Operations Research, INFORMS, vol. 45(2), pages 235-243, April.
    3. Kaas, Rob & Tang, Qihe, 2005. "A large deviation result for aggregate claims with dependent claim occurrences," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 251-259, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Robert, Christian Y. & Segers, Johan, 2008. "Tails of random sums of a heavy-tailed number of light-tailed terms," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 85-92, August.
    2. Fotis Loukissas, 2012. "Precise Large Deviations for Long-Tailed Distributions," Journal of Theoretical Probability, Springer, vol. 25(4), pages 913-924, December.
    3. Yiqing Chen & Kam C. Yuen & Kai W. Ng, 2011. "Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 821-833, December.
    4. Chen, Yiqing & Ng, Kai W., 2007. "The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 415-423, May.
    5. Yang, Haizhong & Sun, Suting, 2013. "Subexponentiality of the product of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2039-2044.
    6. Serguei Foss & Andrew Richards, 2010. "On Sums of Conditionally Independent Subexponential Random Variables," Mathematics of Operations Research, INFORMS, vol. 35(1), pages 102-119, February.
    7. Yang, Yingying & Hu, Shuhe & Wu, Tao, 2011. "The tail probability of the product of dependent random variables from max-domains of attraction," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1876-1882.
    8. Robert, Christian Y., 2013. "Some new classes of stationary max-stable random fields," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1496-1503.
    9. Shen, Xinmei & Lin, Zhengyan, 2008. "Precise large deviations for randomly weighted sums of negatively dependent random variables with consistently varying tails," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3222-3229, December.
    10. Kaas, Rob & Tang, Qihe, 2005. "A large deviation result for aggregate claims with dependent claim occurrences," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 251-259, June.
    11. Yang, Yang & Ignatavičiūtė, Eglė & Šiaulys, Jonas, 2015. "Conditional tail expectation of randomly weighted sums with heavy-tailed distributions," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 20-28.
    12. Jiang, Tao & Wang, Yuebao & Chen, Yang & Xu, Hui, 2015. "Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 45-53.
    13. David Heath & Sidney Resnick & Gennady Samorodnitsky, 1998. "Heavy Tails and Long Range Dependence in On/Off Processes and Associated Fluid Models," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 145-165, February.
    14. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    15. Dan Zhu & Ming Zhou & Chuancun Yin, 2023. "Finite-Time Ruin Probabilities of Bidimensional Risk Models with Correlated Brownian Motions," Mathematics, MDPI, vol. 11(12), pages 1-18, June.
    16. Gao, Qingwu & Liu, Xijun, 2013. "Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1527-1538.
    17. Chen, Yu & Zhang, Weiping, 2007. "Large deviations for random sums of negatively dependent random variables with consistently varying tails," Statistics & Probability Letters, Elsevier, vol. 77(5), pages 530-538, March.
    18. Yang Yang & Xinzhi Wang & Shaoying Chen, 2022. "Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 1221-1236, June.
    19. Peng, Jiangyan & Huang, Jin, 2010. "Ruin probability in a one-sided linear model with constant interest rate," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 662-669, April.
    20. Jiang, Jun & Tang, Qihe, 2011. "The product of two dependent random variables with regularly varying or rapidly varying tails," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 957-961, August.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:math/0703022. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.