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Discrete Mathematics and Applications

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Volume 28, Issue 2


Decomposable branching processes with two types of particles

Vladimir A. Vatutin / Elena E. Dyakonova
Published Online: 2018-04-09 | DOI: https://doi.org/10.1515/dma-2018-0012


A two-type critical decomposable branching process with discrete time is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ2, the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ2 = N or Ξ2 > N.

Keywords: decomposable branching process; total population size; limit theorem

Originally published in Diskretnaya Matematika (2018) 30,№1, 3–18 (in Russian).


  • [1]

    Afanasyev V. I., “Functional limit theorems for the decomposable branching process with two types of particles”, Discrete Math. Appl., 26:2 (2016), 71–88.Google Scholar

  • [2]

    Afanasyev V. I., “On a decomposable branching process with two types of particles”, Proc. Steklov Inst. Math., 294 (2016), 1–12.CrossrefGoogle Scholar

  • [3]

    Afanasyev V. I., “Functional limit theorem for a decomposable branching process with two types of particles”, Math. Notes, 103:3 (2018), 323-335 (in Russian).Google Scholar

  • [4]

    Vatutin V. A., “The structure of decomposable reduced branching processes. I. Finite-dimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662.CrossrefGoogle Scholar

  • [5]

    Vatutin V. A., “The structure of decomposable reduced branching processes. II. Functional limit theorems”, Theory Probab. Appl., 60:1 (2016), 103–119.CrossrefGoogle Scholar

  • [6]

    Vatutin V. A., “A conditional functional limit theorem for decomposable branching processes with two types of particles”, Math. Notes, 101:5 (2017), 778–789.CrossrefGoogle Scholar

  • [7]

    Vatutin V. A., D’yakonova E. E., “Decomposable branching processes with a fixed extinction moment”, Proc. Steklov Inst. Math., 290 (2015), 103–124.CrossrefGoogle Scholar

  • [8]

    Vatutin V. A., Dyakonova E. E., “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192.Google Scholar

  • [9]

    Vatutin V. A., Sagitov S. M., “A decomposable critical branching process with two types of particles”, Proc. Steklov Inst. Math., 177 (1988), 1–19.Google Scholar

  • [10]

    Gnedenko B. V., Kolmogorov A. N., Limit distributions for sums of independent random variables, Addison-Wesley, Reading, London, 1954.Google Scholar

  • [11]

    Foster J., Ney P., “Decomposable critical multi-type branching processes”, Invited paper for Mahalanobis Memorial symposium (Calcutta), Sanhya: the Indian J. Stat., Series A, 38 (1976), 28–37.Google Scholar

  • [12]

    Foster J., Ney P., “Limit laws for decomposable critical branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 46 (1978), 13–43.CrossrefGoogle Scholar

  • [13]

    Ogura Y., “Asymptotic behavior of multitype Galton-Watson processes”, J. Math. Kyoto Univ., 15 (1975), 251–302.CrossrefGoogle Scholar

  • [14]

    Savin A.A., Chistyakov V.P., “Some limit theorems for branching processes with several types of particles”, Theory Probab. Appl., 7:1 (1962), 93–100.CrossrefGoogle Scholar

  • [15]

    Sagitov S. M.,, “Multidimensional limit theorems for a branching process with a single type of particles”, Math. Notes, 42:1 (1987), 597–602.CrossrefGoogle Scholar

  • [16]

    Slack R. S., “A branching process with mean one and possibly infinite variance”, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 9:2 (1968), 139–145.CrossrefGoogle Scholar

  • [17]

    Seneta E., Regularly Varying Functions, Springer-Verlag, Berlin, Heidelberg, New York, 1976.Google Scholar

  • [18]

    Kersting G., “On the height profile of a conditioned Galton-Watson tree”, arXiv:1101.3656 [math.PR], 2011, 24 pages.Google Scholar

  • [19]

    Kolchin V. F., Random Mappings, Optimization Software, Inc., Publication Division, 1986.Google Scholar

  • [20]

    Bertoin J., Levy Processes, Cambrige University Press, Cambridge, 1996.Google Scholar

  • [21]

    Smadi C., Vatutin V. A., “Reduced two-type decomposable critical branching processes with possibly infinite variance”, Markov Process. Related Fields, 22:2 (2016), 311–358.Google Scholar

  • [22]

    Zolotarev V. M., One-dimensional Stable Distributions, Transl. of Math. Monographs, vol. 65, Amer. Math. Soc., Providence, 1986.Google Scholar

  • [23]

    Zubkov A. M., “The limit behaviour of decomposable critical branching processes with two types of particles”, Theory Probab. Appl., 27:2 (1983), 235–237.CrossrefGoogle Scholar

  • [24]

    Feller W., An Introduction to Probability Theory and its Applications. Vol II, John Wiley & Sons, Inc., New York, London, Sydney, Toronto, 1971.Google Scholar

About the article

Received: 2017-10-20

Published Online: 2018-04-09

Published in Print: 2018-04-25

Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-50-00005

This work was supported by the Russian Science Foundation under grant no. 14-50-00005.

Citation Information: Discrete Mathematics and Applications, Volume 28, Issue 2, Pages 119–130, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265, DOI: https://doi.org/10.1515/dma-2018-0012.

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