IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v164y2018icp40-53.html
   My bibliography  Save this article

On the joint tail behavior of randomly weighted sums of heavy-tailed random variables

Author

Listed:
  • Li, Jinzhu

Abstract

We focus on the joint tail behavior of randomly weighted sums Sn=U1X1+⋯+UnXn and Tm=V1Y1+⋯+VmYm. The vectors of primary random variables (X1,Y1), (X2,Y2),… are assumed to be independent with dominatedly varying marginal distributions, and the dependence within each pair (Xi,Yi) satisfies a condition called strong asymptotic independence. The random weights U1, V1,… are non-negative and arbitrarily dependent, but they are independent of the primary random variables. Under suitable conditions, we obtain asymptotic expansions for the joint tails of (Sn,Tm) with fixed positive integers n and m, and (SN,TM) with integer-valued random variables N and M that are independent of the primary random variables. When the marginal distributions of the primary random variables are extended regularly varying, the result is proved to hold uniformly for any n and m under stronger conditions. Our results rely critically on moment conditions that are generally easy to check.

Suggested Citation

  • Li, Jinzhu, 2018. "On the joint tail behavior of randomly weighted sums of heavy-tailed random variables," Journal of Multivariate Analysis, Elsevier, vol. 164(C), pages 40-53.
    • Handle: RePEc:eee:jmvana:v:164:y:2018:i:c:p:40-53
    DOI: 10.1016/j.jmva.2017.10.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X17306553
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2017.10.008?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Nyrhinen, Harri, 1999. "On the ruin probabilities in a general economic environment," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 319-330, October.
    2. Nyrhinen, Harri, 2001. "Finite and infinite time ruin probabilities in a stochastic economic environment," Stochastic Processes and their Applications, Elsevier, vol. 92(2), pages 265-285, April.
    3. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    4. Zhang, Yi & Shen, Xinmei & Weng, Chengguo, 2009. "Approximation of the tail probability of randomly weighted sums and applications," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 655-675, February.
    5. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 299-325, December.
    6. Bolancé, Catalina & Bahraoui, Zuhair & Artís, Manuel, 2014. "Quantifying the risk using copulae with nonparametric marginals," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 46-56.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Gajek, Lesław & Krajewska, Elżbieta, 2020. "Approximating sums of products of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 164(C).
    2. Richards, Jordan & Tawn, Jonathan A., 2022. "On the tail behaviour of aggregated random variables," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    3. Chen, Yiqing, 2020. "A Kesten-type bound for sums of randomly weighted subexponential random variables," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Lin, Jianxi, 2019. "Second order tail approximation for the maxima of randomly weighted sums with applications to ruin theory and numerical examples," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 37-47.

    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. Chen Yu & Zhang Weiping & Liu Jie, 2010. "Asymptotic Tail Probability of Randomly Weighted Sum of Dependent Heavy-Tailed Random Variables," Asia-Pacific Journal of Risk and Insurance, De Gruyter, vol. 4(2), pages 1-11, July.
    2. Sun, Ying & Wei, Li, 2014. "The finite-time ruin probability with heavy-tailed and dependent insurance and financial risks," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 178-183.
    3. Chen, Yiqing & Liu, Jiajun & Liu, Fei, 2015. "Ruin with insurance and financial risks following the least risky FGM dependence structure," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 98-106.
    4. Yang, Yang & Jiang, Tao & Wang, Kaiyong & Yuen, Kam C., 2020. "Interplay of financial and insurance risks in dependent discrete-time risk models," Statistics & Probability Letters, Elsevier, vol. 162(C).
    5. Eckert, Johanna & Gatzert, Nadine, 2018. "Risk- and value-based management for non-life insurers under solvency constraints," European Journal of Operational Research, Elsevier, vol. 266(2), pages 761-774.
    6. 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.
    7. Nyrhinen, Harri, 2007. "Convex large deviation rate functions under mixtures of linear transformations, with an application to ruin theory," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 947-959, July.
    8. Cai, Jun & Dickson, David C.M., 2004. "Ruin probabilities with a Markov chain interest model," Insurance: Mathematics and Economics, Elsevier, vol. 35(3), pages 513-525, December.
    9. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    10. Jaakko Lehtomaa, 2015. "Asymptotic Behaviour of Ruin Probabilities in a General Discrete Risk Model Using Moment Indices," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1380-1405, December.
    11. Yiqing Chen & Jiajun Liu & Yang Yang, 2023. "Ruin under Light-Tailed or Moderately Heavy-Tailed Insurance Risks Interplayed with Financial Risks," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    12. Tang, Qihe & Vernic, Raluca, 2007. "The impact on ruin probabilities of the association structure among financial risks," Statistics & Probability Letters, Elsevier, vol. 77(14), pages 1522-1525, August.
    13. Yuchao Dong & Jérôme Spielmann, 2020. "Weak Limits of Random Coefficient Autoregressive Processes and their Application in Ruin Theory," Post-Print hal-02170829, HAL.
    14. Xin-mei Shen & Zheng-yan Lin & Yi Zhang, 2009. "Uniform Estimate for Maximum of Randomly Weighted Sums with Applications to Ruin Theory," Methodology and Computing in Applied Probability, Springer, vol. 11(4), pages 669-685, December.
    15. Zhang, Yi & Shen, Xinmei & Weng, Chengguo, 2009. "Approximation of the tail probability of randomly weighted sums and applications," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 655-675, February.
    16. Qu, Zhihui & Chen, Yu, 2013. "Approximations of the tail probability of the product of dependent extremal random variables and applications," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 169-178.
    17. Jaunė, Eglė & Šiaulys, Jonas, 2022. "Asymptotic risk decomposition for regularly varying distributions with tail dependence," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    18. Yuchao Dong & J'er^ome Spielmann, 2019. "Weak Limits of Random Coefficient Autoregressive Processes and their Application in Ruin Theory," Papers 1907.01828, arXiv.org, revised Feb 2020.
    19. Yuchao Dong & Jérôme Spielmann, 2019. "Weak Limits of Random Coefficient Autoregressive Processes and their Application in Ruin Theory," Working Papers hal-02170829, HAL.
    20. Yang Yang & Shuang Liu & Kam Chuen Yuen, 2022. "Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2600-2621, December.

    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:eee:jmvana:v:164:y:2018:i:c:p:40-53. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.