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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.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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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
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[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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