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The life spans of a Bellman-Harris branching process with immigration

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Abstract

One considers two schemes of the Bellman-Harris process with immigration when a) the lifetime of the particles is an integral-valued random variable and the immigration is defined by a sequence of independent random variables; b) the distribution of the lifetime of the particles is nonlattice and the immigration is a process with continuous time. One investigates the properties of the life spans of such processes. The results obtained here are a generalization to the case of Bellman-Harris processes of the results of A. M. Zubkov, obtained for Markov branching processes. For the proof one makes use in an essential manner of the known inequalities of Goldstein, estimating the generating function of the Bellman-Harris process in terms of the generating functions of the imbedded Galton-Watson process.

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Literature cited

  1. G. P. Klimov, Stochastic Queueing Systems [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  2. B. A. Sevast'yanov, Branching Processes [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  3. A. G. Pakes, “On the critical Galton-Watson process with immigration,” J. Austral. Math. Soc.,12, No. 4, 476–482 (1971).

    Google Scholar 

  4. K. B. Athreya and P. E. Ney, Branching Processes, Springer, New York (1972).

    Google Scholar 

  5. A. M. Zubkov, “The life spans of a branching process with immigration,” Teor. Veroyatn. Primen.,17, No. 1, 179–188 (1972).

    Google Scholar 

  6. V. A. Vatutin, “Discrete limit distributions of the number of particles in critical Bellman-Harris branching processes,” Teor. Veroyatn. Primen.,22, No. 1, 150–155 (1977).

    Google Scholar 

  7. V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975).

    Google Scholar 

  8. W. Feller, An Introduction to Probability Theory and Its Applications, Vols. I, II, Wiley, New York (1968 and 1966).

    Google Scholar 

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Authors
  1. I. S. Badalbaev
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  2. A. Mashrabbaev
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Additional information

Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 60–82, 1986.

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Badalbaev, I.S., Mashrabbaev, A. The life spans of a Bellman-Harris branching process with immigration. J Math Sci 38, 2198–2210 (1987). https://doi.org/10.1007/BF01093821

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  • DOI: https://doi.org/10.1007/BF01093821

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