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

*(English. Russian original)*Zbl 0668.60076
J. Sov. Math. 38, No. 5, 2198-2210 (1987); translation from Probability distributions and mathematical statistics, Collect. Artic., Tashkent 1986, 60-82 (1986).

[For the entire collection see Zbl 0626.00025.]

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 [Theory Probab. Appl. 17, 174-183 (1972; Zbl 0267.60084); translation from Teor. Veroyatn. Primen. 17, 179-188 (1972)] obtained for Markov branching processes. For the proof one makes use in an essential manner of the well-known Goldstein inequalities estimating the generating function of the Bellman-Harris process in terms of the generating functions of the imbedded Galton- Watson process.

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 [Theory Probab. Appl. 17, 174-183 (1972; Zbl 0267.60084); translation from Teor. Veroyatn. Primen. 17, 179-188 (1972)] obtained for Markov branching processes. For the proof one makes use in an essential manner of the well-known Goldstein inequalities estimating the generating function of the Bellman-Harris process in terms of the generating functions of the imbedded Galton- Watson process.

##### MSC:

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

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\textit{I. S. Badalbaev} and \textit{A. Mashrabbaev}, J. Sov. Math. 38, No. 5, 2198--2210 (1987; Zbl 0668.60076); translation from Probability distributions and mathematical statistics, Collect. Artic., Tashkent 1986, 60--82 (1986)

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##### References:

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

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

[3] | A. G. Pakes, ?On the critical Galton-Watson process with immigration,? J. Austral. Math. Soc.,12, No. 4, 476?482 (1971). · Zbl 0249.60045 · doi:10.1017/S1446788700010375 |

[4] | K. B. Athreya and P. E. Ney, Branching Processes, Springer, New York (1972). · Zbl 0259.60002 |

[5] | A. M. Zubkov, ?The life spans of a branching process with immigration,? Teor. Veroyatn. Primen.,17, No. 1, 179?188 (1972). · Zbl 0267.60084 |

[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). · Zbl 0391.60082 |

[7] | V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975). · Zbl 0322.60043 |

[8] | W. Feller, An Introduction to Probability Theory and Its Applications, Vols. I, II, Wiley, New York (1968 and 1966). · Zbl 0138.10207 |

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