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Discrete Mathematics and Applications

Editor-in-Chief: Zubkov, Andrei

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Volume 28, Issue 5


Reduced critical Bellman–Harris branching processes for small populations

Vladimir A. Vatutin / Wenming Hong
  • School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University Beijing, China
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/ Yao Ji
  • School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University Beijing, China
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Published Online: 2018-10-26 | DOI: https://doi.org/10.1515/dma-2018-0028


A critical Bellman-Harris branching process {Z(t),t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z(t) ≤ φ(t), where either φ(t) = o(t) as t → ∞ or φ(t) = at,a>0, we study the structure of the process where Z(s,t) is the number of particles in the initial process at moment s which either survive up to moment t or have a positive number of descendants at this moment.

Keywords: Bellman-Harris branching process; reduced process; conditional limit theorem

Originally published in Diskretnaya Matematika (2018) 30,No 3, 25–39 (in Russian).


The work of V.A. Vatutin was supported by the Russian Science Foundation under grant no. 14-50-00005, the work of Wenming Hong and Yao Ji was supported by Natural Science Foundation of China under grants 11531001 and 11626245.


  • [1]

    Athreya K. B., “Coalescence in the recent past in rapidly growing populations”, Stoch. Proc. and their Appl., 122:11 (2012) 3757–3766Web of ScienceGoogle Scholar

  • [2]

    Athreya K. B., “Coalescence in critical and subcritical Galton-Watson branching processes”, J. Appl. Probab., 49:3 (2012) 627–638Google Scholar

  • [3]

    Durrett R., “The genealogy of critical branching processes”, Stoch. Proc. and their Appl. 8:1 (1978) 101–116Google Scholar

  • [4]

    Fleischmann K., Prehn U., “Ein Grenzfersatz fur subkritische Verzweigungsprozesse mit eindlich vielen Typen von Teilchen”, Math. Nachr. 64 (1974),233–241Google Scholar

  • [5]

    Fleischmann K., Siegmund-Schultze R., “The structure of reduced critical Galton-Watson processes”, Math. Nachr. 79 (1977) 357–362Google Scholar

  • [6]

    Goldstein M., “Critical age-dependent branching processes: single and multitype”, Z. Wahrscheinlichkeitstheor. verw. Geb. 17:2 (1971) 74–78Google Scholar

  • [7]

    Harris S. C., Johnston S. G. G., Roberts M. I., “The coalescent structure of continuous-time Galton-Watson trees”, 2017 https://arxiv.org/pdf/1703.00299.pdfGoogle Scholar

  • [8]

    Johnston S. G. G., “Coalescence in supercritical and subcritical continuous-time Galton-Watson trees”, 2017 https://arxiv.org/pdf/1709.008500v1.pdfGoogle Scholar

  • [9]

    Lambert A., “Coalescence times for the branching process”, Adv. Appl. Probab. 35:4 (2003) 1071–1089Google Scholar

  • [10]

    Le V., “Coalescence times for the Bienaym\'e-Galton-Watson process”, J. Appl. Probab. 51:1 (2014) 209–218Google Scholar

  • [11]

    Liu M., Vatutin V., “Reduced processes for small populations”, Theory Probab. Appl. 63:4 (2018) (toappear)Google Scholar

  • [12]

    Sagitov S. M., “Reduced multitype critical Bellman–Harris branching process”, Theory Probab. Appl. 30:4 (1986) 783–796Google Scholar

  • [13]

    Topchii V. A., “A local limit theorem for critical Bellman–Harris processes with discrete time”, In: Limit theorems of probability theory and related questions, Trudy Inst. Mat., Nauka, Sibirsk. Otdel., Novosibirsk 1 (1982) 197–122 in RussianGoogle Scholar

  • [14]

    Vatutin V. A., “Discrete limit distributions for the number of particles in the critical Bellman–Harris branching processes”, Theory Probab. Appl. 22:1 (1977) 146–152Google Scholar

  • [15]

    Vatutin V. A., “Distance to the nearest common ancestor in Bellman–Harris branching processes”, Math. Notes 25:5 (1979) 378–382Google Scholar

  • [16]

    Vatutin V. A., “A local limit theorem for critical Bellman–Harris branching processes”, Proc. Steklov Inst. Math. 158(1983) 9–31Google Scholar

  • [17]

    Zubkov A.M., “Limit distributions of the distance to the nearest common ancestor”, Theory Probab. Appl. 20:3 (1975) 602–612Google Scholar

About the article

Received: 2018-05-17

Published Online: 2018-10-26

Published in Print: 2018-10-25

Citation Information: Discrete Mathematics and Applications, Volume 28, Issue 5, Pages 319–330, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265, DOI: https://doi.org/10.1515/dma-2018-0028.

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