IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i3p271-d489517.html
   My bibliography  Save this article

Characterization of Probability Distributions via Functional Equations of Power-Mixture Type

Author

Listed:
  • Chin-Yuan Hu

    (National Changhua University of Education, Changhua 50058, Taiwan)

  • Gwo Dong Lin

    (Social and Data Science Research Center, Hwa-Kang Xing-Ye Foundation, Taipei 10659, Taiwan
    Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan)

  • Jordan M. Stoyanov

    (Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria)

Abstract

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [ 0 , ∞ ) . These equations arise when studying distributional equations of the type Z = d X + T Z , where the random variable T ≥ 0 has known distribution, while the distribution of the random variable Z ≥ 0 is a transformation of that of X ≥ 0 , and we want to find the distribution of X . We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results that are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.

Suggested Citation

  • Chin-Yuan Hu & Gwo Dong Lin & Jordan M. Stoyanov, 2021. "Characterization of Probability Distributions via Functional Equations of Power-Mixture Type," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:271-:d:489517
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/3/271/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/3/271/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Lin, Gwo Dong & Hu, Chin-Yuan, 2001. "Characterizations of distributions via the stochastic ordering property of random linear forms," Statistics & Probability Letters, Elsevier, vol. 51(1), pages 93-99, January.
    2. A. E. Eckberg, 1977. "Sharp Bounds on Laplace-Stieltjes Transforms, with Applications to Various Queueing Problems," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 135-142, May.
    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. Wouter van Eekelen & Dick den Hertog & Johan S.H. van Leeuwaarden, 2022. "MAD Dispersion Measure Makes Extremal Queue Analysis Simple," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1681-1692, May.
    2. Freeman, Mark C. & Groom, Ben, 2016. "How certain are we about the certainty-equivalent long term social discount rate?," Journal of Environmental Economics and Management, Elsevier, vol. 79(C), pages 152-168.
    3. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.
    4. Yan Chen & Ward Whitt, 2020. "Algorithms for the upper bound mean waiting time in the GI/GI/1 queue," Queueing Systems: Theory and Applications, Springer, vol. 94(3), pages 327-356, April.
    5. Carlos Chaves & Abhijit Gosavi, 2022. "On general multi-server queues with non-poisson arrivals and medium traffic: a new approximation and a COVID-19 ventilator case study," Operational Research, Springer, vol. 22(5), pages 5205-5229, November.
    6. Yan Chen & Ward Whitt, 2021. "Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 101-124, February.
    7. Iosif Pinelis, 2016. "On the extreme points of moments sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 83(3), pages 325-349, June.
    8. Hansjörg Albrecher & José Carlos Araujo-Acuna, 2022. "On The Randomized Schmitter Problem," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 515-535, June.

    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:gam:jmathe:v:9:y:2021:i:3:p:271-:d:489517. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.