IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v127y2017i9p3014-3041.html
   My bibliography  Save this article

Thin tails of fixed points of the nonhomogeneous smoothing transform

Author

Listed:
  • Alsmeyer, Gerold
  • Dyszewski, Piotr

Abstract

For a given random sequence (C,T1,T2,…), the smoothing transform S maps the law of a real random variable X to the law of ∑k≥1TkXk+C, where X1,X2,… are independent copies of X and also independent of (C,T1,T2,…). This law is a fixed point of S if X=d∑k≥1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,…) such that the canonical solution exhibits right and/or left Poissonian tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.

Suggested Citation

  • Alsmeyer, Gerold & Dyszewski, Piotr, 2017. "Thin tails of fixed points of the nonhomogeneous smoothing transform," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 3014-3041.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:3014-3041
    DOI: 10.1016/j.spa.2017.01.008
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917300236
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.01.008?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. Alsmeyer, Gerold & Rösler, Uwe, 2003. "The best constant in the Topchii-Vatutin inequality for martingales," Statistics & Probability Letters, Elsevier, vol. 65(3), pages 199-206, November.
    2. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, 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. Daley, D.J. & Goldie, Charles M., 2006. "The moment index of minima (II)," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 831-837, April.
    2. 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.
    3. 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.
    4. Yuan, Meng & Lu, Dawei, 2022. "Precise large deviation for sums of sub-exponential claims with the m-dependent semi-Markov type structure," Statistics & Probability Letters, Elsevier, vol. 185(C).
    5. Royi Jacobovic & Nikki Levering & Onno Boxma, 2023. "Externalities in the M/G/1 queue: LCFS-PR versus FCFS," Queueing Systems: Theory and Applications, Springer, vol. 104(3), pages 239-267, August.
    6. Kamphorst, Bart & Zwart, Bert, 2019. "Uniform asymptotics for compound Poisson processes with regularly varying jumps and vanishing drift," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 572-603.
    7. Yoni Nazarathy & Zbigniew Palmowski, 2022. "On busy periods of the critical GI/G/1 queue and BRAVO," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 219-225, October.
    8. Shen, Xinmei & Xu, Menghao & Mills, Ebenezer Fiifi Emire Atta, 2016. "Precise large deviation results for sums of sub-exponential claims in a size-dependent renewal risk model," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 6-13.
    9. Lu, Dawei, 2011. "Lower and upper bounds of large deviation for sums of subexponential claims in a multi-risk model," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1911-1919.
    10. G. Alsmeyer & U. Rösler, 2004. "On the Existence of φ-Moments of the Limit of a Normalized Supercritical Galton–Watson Process," Journal of Theoretical Probability, Springer, vol. 17(4), pages 905-928, October.
    11. Lu, Dawei & Zhang, Bin, 2016. "Some asymptotic results of the ruin probabilities in a two-dimensional renewal risk model with some strongly subexponential claims," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 20-29.
    12. Leipus, Remigijus & Siaulys, Jonas, 2007. "Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes," Insurance: Mathematics and Economics, Elsevier, vol. 40(3), pages 498-508, May.
    13. Baltrunas, Aleksandras & Leipus, Remigijus & Siaulys, Jonas, 2008. "Precise large deviation results for the total claim amount under subexponential claim sizes," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1206-1214, August.

    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:spapps:v:127:y:2017:i:9:p:3014-3041. 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/505572/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.