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On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

  • Assen Tchorbadjieff ORCID logo EMAIL logo , Penka Mayster and Anthony G. Pakes
Published/Copyright: March 26, 2024
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Stochastics and Quality Control
From the journal Stochastics and Quality Control Volume 39 Issue 1

Abstract

The probability generating function (pgf) B ⁢ ( s ) of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 and B ′ ⁢ ( 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 ) 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.

Keywords: Subcritical Markov Branching Process; Conditional Limiting Law; Log-Series Law; Borel Law; Sibuya Law; Lambert and Cayley Functions; Lagrange Reversion; Infinite Divisibility
MSC 2020: 60J80; 60K05; 11B83

Acknowledgements

We thank the referee for directing us to the interesting paper [13].

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Received: 2023-12-01
Revised: 2024-02-27
Accepted: 2024-03-01
Published Online: 2024-03-26
Published in Print: 2024-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. 80th Jubilee of Professor Nikolay Yanev
  3. My 55 Years in Stochastics
  4. On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
  5. Galton–Watson Theta-Processes in a Varying Environment
  6. Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
  7. Critical Multitype Branching Processes with Random Migration
  8. Branching Processes Under Nonstandard Conditions
https://doi.org/10.1515/eqc-2023-0043
Keywords for this article
Subcritical Markov Branching Process; Conditional Limiting Law; Log-Series Law; Borel Law; Sibuya Law; Lambert and Cayley Functions; Lagrange Reversion; Infinite Divisibility

Articles in the same Issue

  1. Frontmatter
  2. 80th Jubilee of Professor Nikolay Yanev
  3. My 55 Years in Stochastics
  4. On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
  5. Galton–Watson Theta-Processes in a Varying Environment
  6. Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
  7. Critical Multitype Branching Processes with Random Migration
  8. Branching Processes Under Nonstandard Conditions
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