IDEAS home Printed from https://ideas.repec.org/a/spr/sankha/v86y2024i2d10.1007_s13171-024-00346-w.html
   My bibliography  Save this article

On the Convolution of Scaled Sibuya Distributions

Author

Listed:
  • Nadjib Bouzar

    (University of Indianapolis)

Abstract

We introduce a new heavy tailed distribution on $$\mathbb {Z}_+$$ Z + that arises as the infinite convolution of scaled Sibuya distributions. We provide closed form expressions for its probability mass function, its cumulative distribution function, and its probability generating function. We interpret our main results in terms of the weak convergence of partial sums of a binomially thinned sequence of i.i.d. random variables with a common scaled Sibuya distribution. Properties of infinite divisibility and discrete self-decomposability of the new distribution are also discussed. As an application, we briefly describe an integer-valued autoregressive process of order one with a scaled Sibuya innovation sequence. Finally, we discuss some partial extensions of our results to the case of the generalized Sibuya distribution introduced by Kozubowski and Podgórski., Ann. of the Inst. Statist. Math., 70(4), 855-887., 2018.

Suggested Citation

  • Nadjib Bouzar, 2024. "On the Convolution of Scaled Sibuya Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 699-720, August.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:2:d:10.1007_s13171-024-00346-w
    DOI: 10.1007/s13171-024-00346-w
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13171-024-00346-w
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13171-024-00346-w?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. Tomasz J. Kozubowski & Krzysztof Podgórski, 2018. "A generalized Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 855-887, August.
    2. Christoph, Gerd & Schreiber, Karina, 2000. "Scaled Sibuya distribution and discrete self-decomposability," Statistics & Probability Letters, Elsevier, vol. 48(2), pages 181-187, 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. Peter Kern & Svenja Lage, 2023. "On Self-Similar Bernstein Functions and Corresponding Generalized Fractional Derivatives," Journal of Theoretical Probability, Springer, vol. 36(1), pages 348-371, March.
    2. Thierry E. Huillet, 2022. "Chance Mechanisms Involving Sibuya Distribution and its Relatives," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 722-764, November.
    3. Nadjib Bouzar & K. Jayakumar, 2008. "Time series with discrete semistable marginals," Statistical Papers, Springer, vol. 49(4), pages 619-635, October.
    4. Nadjib Bouzar, 2008. "The semi-Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 459-464, June.
    5. Rodrigues, Josemar & Balakrishnan, N. & Cordeiro, Gauss M. & de Castro, Mário, 2011. "A unified view on lifetime distributions arising from selection mechanisms," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3311-3319, December.
    6. N. Bouzar & S. Satheesh, 2008. "Comments on a-decomposability," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 243-252.
    7. Thomas M. Michelitsch & Federico Polito & Alejandro P. Riascos, 2023. "Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times," Mathematics, MDPI, vol. 11(2), pages 1-20, January.
    8. Thierry E. Huillet, 2020. "On New Mechanisms Leading to Heavy-Tailed Distributions Related to the Ones Of Yule-Simon," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(1), pages 321-344, March.
    9. Balakrishnan, N. & Jones, M.C., 2022. "Closure of beta and Dirichlet distributions under discrete mixing," Statistics & Probability Letters, Elsevier, vol. 188(C).
    10. Tomasz J. Kozubowski & Krzysztof Podgórski, 2018. "A generalized Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 855-887, August.
    11. Jones, M.C. & Balakrishnan, N., 2021. "Simple functions of independent beta random variables that follow beta distributions," Statistics & Probability Letters, Elsevier, vol. 170(C).
    12. Nadjib Bouzar, 2008. "Semi-self-decomposable distributions on Z +," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(4), pages 901-917, December.
    13. Zhu, Rong & Joe, Harry, 2009. "Modelling heavy-tailed count data using a generalised Poisson-inverse Gaussian family," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1695-1703, August.
    14. Soltani, A.R. & Shirvani, A. & Alqallaf, F., 2009. "A class of discrete distributions induced by stable laws," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1608-1614, 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:spr:sankha:v:86:y:2024:i:2:d:10.1007_s13171-024-00346-w. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.