Startseite On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

  • Assen Tchorbadjieff ORCID logo EMAIL logo , Penka Mayster und Anthony G. Pakes
Veröffentlicht/Copyright: 26. März 2024
Veröffentlichen auch Sie bei De Gruyter Brill
Manuskript einreichen Informationen für Autor*innen
Stochastics and Quality Control
Aus der Zeitschrift Stochastics and Quality Control Band 39 Heft 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].

References

[1] S. Asmussen and H. Hering, Branching Processes, Progr. Probab. Stat. 3, Birkhäuser, Boston, 1983. 10.1007/978-1-4615-8155-0Suche in Google Scholar

[2] K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren Math. Wiss. 196, Springer, New Yorkg, 1972. 10.1007/978-3-642-65371-1Suche in Google Scholar

[3] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl. 27, Cambridge University, Cambridge, 1987. 10.1017/CBO9780511721434Suche in Google Scholar

[4] P. C. Consul and F. Famoye, Lagrangian Probability Distributions, Birkhäuser, Boston, 2006. Suche in Google Scholar

[5] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996), no. 4, 329–359. 10.1007/BF02124750Suche in Google Scholar

[6] N. S. Goel and N. Richter-Dyn, Stochastic Models in Biology, Academic Press, New York, 1974. Suche in Google Scholar

[7] T. E. Harris, The Theory of Branching Processes, Grundlehren Math. Wiss. 119, Springer, Berlin, 1963. 10.1007/978-3-642-51866-9Suche in Google Scholar

[8] N. L. Johnson, A. W. Kemp and S. Kotz, Univariate Discrete Distributions, 3rd ed., Wiley Ser. Probab. Stat., Wiley-Interscience, Hoboken, 2005. 10.1002/0471715816Suche in Google Scholar

[9] P. Mayster and A. Tchorbadjieff, Extended Sibuya distribution in subcritical Markov branching processes, C. R. Acad. Bulgare Sci. 76 (2023), no. 4, 517–524. 10.7546/CRABS.2023.04.02Suche in Google Scholar

[10] F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions, Cambridge University, Cambridge, 2010. Suche in Google Scholar

[11] A. G. Pakes, On the recognition and structure of probability generating functions, Classical and Modern Branching Processes (Minneapolis 1994), IMA Vol. Math. Appl. 84, Springer, New York (1997), 263–284. 10.1007/978-1-4612-1862-3_21Suche in Google Scholar

[12] H. Rubin and D. Vere-Jones, Domains of attraction for the subcritical Galton–Watson branching process, J. Appl. Probab. 5 (1968), 216–219. 10.1017/S0021900200032411Suche in Google Scholar

[13] S. Sagitov and A. Lindo, A special family of Galton–Watson processes with explosions, Branching Processes and Their Applications, Lect. Notes Stat. 219, Springer, Cham (2016), 237–254. 10.1007/978-3-319-31641-3_14Suche in Google Scholar

[14] F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Monogr. Textb. Pure. Appl. Math. 259, Marcel Dekker, New York, 2004. 10.1201/9780203014127Suche in Google Scholar

[15] A. Tchorbadjieff and P. Mayster, Geometric branching reproduction Markov processes, Mod. Stoch. Theory Appl. 7 (2020), no. 4, 357–378. 10.15559/20-VMSTA163Suche in Google Scholar
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

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Subcritical Markov Branching Process; Conditional Limiting Law; Log-Series Law; Borel Law; Sibuya Law; Lambert and Cayley Functions; Lagrange Reversion; Infinite Divisibility

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 24.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/eqc-2023-0043/html?lang=de
Button zum nach oben scrollen