IDEAS home Printed from https://ideas.repec.org/a/bpj/ecqcon/v39y2024i1p9-23n1001.html
   My bibliography  Save this article

On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Author

Listed:
  • Tchorbadjieff Assen

    (Institute of Mathematics & Informatics, [87189]Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bloc 8, 1113 Sofia, Bulgaria)

  • Mayster Penka

    (Institute of Mathematics & Informatics, [87189]Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bloc 8, 1113 Sofia, Bulgaria)

  • Pakes Anthony G.

    (Department of Mathematics & Statistics, [550237]University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia)

Abstract

The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Suggested Citation

  • Tchorbadjieff Assen & Mayster Penka & Pakes Anthony G., 2024. "On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law," Stochastics and Quality Control, De Gruyter, vol. 39(1), pages 9-23.
    • Handle: RePEc:bpj:ecqcon:v:39:y:2024:i:1:p:9-23:n:1001
    DOI: 10.1515/eqc-2023-0043
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/eqc-2023-0043
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/eqc-2023-0043?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.

    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:bpj:ecqcon:v:39:y:2024:i:1:p:9-23:n:1001. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.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.