IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v55y2001i1p89-97.html
   My bibliography  Save this article

Some estimates of geometric sums

Author

Listed:
  • Bon, Jean-Louis
  • Kalashnikov, Vladimir

Abstract

The paper is devoted to analysis of geometric convolutions emerging in various fields of applied probability and, in particular, in reliability. The problem of bounding the distribution of such sums has been the subject of numerous works for last 20 years. Various bounds were proposed but their accuracy was sometimes not satisfactory for applications to highly reliable systems especially in the case of relatively small values of the time argument. Using truncation arguments, we propose new two-sided inequalities improving some known bounds.

Suggested Citation

  • Bon, Jean-Louis & Kalashnikov, Vladimir, 2001. "Some estimates of geometric sums," Statistics & Probability Letters, Elsevier, vol. 55(1), pages 89-97, November.
  • Handle: RePEc:eee:stapro:v:55:y:2001:i:1:p:89-97
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167-7152(01)00136-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Embrechts, P. & Veraverbeke, N., 1982. "Estimates for the probability of ruin with special emphasis on the possibility of large claims," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 55-72, January.
    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. Cai, Jun & Willmot, Gordon E., 2005. "Monotonicity and aging properties of random sums," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 381-392, July.
    2. Chadjiconstantinidis, Stathis & Xenos, Panos, 2022. "Refinements of bounds for tails of compound distributions and ruin probabilities," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    3. Mathieu Emily & Pierre Casez & Olivier François, 2009. "Risk Assessment for Hospital‐Acquired Diseases: A Risk‐Theory Approach," Risk Analysis, John Wiley & Sons, vol. 29(4), pages 565-575, April.
    4. Mercier, Sophie, 2007. "Discrete random bounds for general random variables and applications to reliability," European Journal of Operational Research, Elsevier, vol. 177(1), pages 378-405, February.

    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. Furrer, Hansjorg & Michna, Zbigniew & Weron, Aleksander, 1997. "Stable Lévy motion approximation in collective risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 20(2), pages 97-114, September.
    2. Grandell, Jan, 2000. "Simple approximations of ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 26(2-3), pages 157-173, May.
    3. S. Pitts, 1994. "Nonparametric estimation of compound distributions with applications in insurance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(3), pages 537-555, September.
    4. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    5. Julien Trufin & Stéphane Loisel, 2013. "Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments," Post-Print hal-00426790, HAL.
    6. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
    7. Yuen, Kam C. & Wang, Guojing & Ng, Kai W., 2004. "Ruin probabilities for a risk process with stochastic return on investments," Stochastic Processes and their Applications, Elsevier, vol. 110(2), pages 259-274, April.
    8. Florin Avram & Romain Biard & Christophe Dutang & Stéphane Loisel & Landy Rabehasaina, 2014. "A survey of some recent results on Risk Theory," Post-Print hal-01616178, HAL.
    9. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    10. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    11. Konstantinides, Dimitrios & Tang, Qihe & Tsitsiashvili, Gurami, 2002. "Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails," Insurance: Mathematics and Economics, Elsevier, vol. 31(3), pages 447-460, December.
    12. Vaios Dermitzakis & Konstadinos Politis, 2011. "Asymptotics for the Moments of the Time to Ruin for the Compound Poisson Model Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 13(4), pages 749-761, December.
    13. J. Cerda-Hernandez & A. Sikov, 2022. "Optimal investment strategy to maximize the expected utility of an insurance company under Cramer Lundberg dynamic," Papers 2207.02947, arXiv.org.
    14. M. S. Sgibnev, 1998. "On the Asymptotic Behavior of the Harmonic Renewal Measure," Journal of Theoretical Probability, Springer, vol. 11(2), pages 371-382, April.
    15. Griffin, Philip S. & Maller, Ross A. & Schaik, Kees van, 2012. "Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 382-392.
    16. 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.
    17. Conti, Pier Luigi, 2005. "A nonparametric sequential test with power 1 for the ruin probability in some risk models," Statistics & Probability Letters, Elsevier, vol. 72(4), pages 333-343, May.
    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. Toshiro Watanabe & Kouji Yamamuro, 2010. "Local Subexponentiality and Self-decomposability," Journal of Theoretical Probability, Springer, vol. 23(4), pages 1039-1067, December.
    20. Ramsay, Colin M., 2003. "A solution to the ruin problem for Pareto distributions," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 109-116, August.
    21. Wang, Dingcheng & Chen, Pingyan & Su, Chun, 2007. "The supremum of random walk with negatively associated and heavy-tailed steps," Statistics & Probability Letters, Elsevier, vol. 77(13), pages 1403-1412, July.

    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:stapro:v:55:y:2001:i:1:p:89-97. 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.