Literature cited
B. A. Sevast'yanov, Branching Processes [in Russian], Nauka, Moscow (1971).
T. E. Harris, The Theory of Branching Processes, Springer, Berlin (1963).
K. B. Athreya and P. E. Ney, Branching Processes, Springer, New York (1972).
J. Lamperti, “The limit of a sequence of branching processes,” Z. Wahrsch. Verw. Gebiete7, 7, 271–288 (1967).
A. Grimvall, “On the convergence of sequences of branching processes,” Ann. Probab.,2, No. 6, 1027–1045 (1974).
S. A. Aliev and V. M. Shurenkov, “Transition phenomena and the convergence of Galton-Watson processes to Jirina processes,” Teor. Veroyatn. Primen.,27, No. 3, 443–455 (1982).
V. M. Shurenkov, “Two limit theorems for critical branching processes,” Teor. Veroyatn. Primen.,21, No. 3, 389–394 (1976).
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).
A. L. Yakymiv, “The asymptotic behavior of the probabilities of the extensions of critical Bellman-Harris processes,” Tr. Mat. Inst. Akad. Nauk SSSR,177, 177–205 (1986).
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading (1954).
R. S. Slack, “A branching process with mean one and possibly infinite variance,” Z. Wahrsch. Verw. Gebiete,9, No. 2, 139–145 (1968).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York (1966).
E. Seneta, Regularly Varying Functions, Lecture Notes in Math., No. 508, Springer, Berlin (1976).
S. G. Tkachuk, “Theorems on large deviations in the case of distributions with regularly varying tails,” in: Random Processes and Statistical Inference, No. 5 [in Russian], Fan, Tashkent (1975), pp. 164–174.
A. N. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972).
V. A. Vatutin and S. M. Sagitov, “Decomposable branching processes with two types of particles,” Trudy Mat. Inst. Akad. Nauk SSSR,177, 3–20 (1986).
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Translated from Matematicheskie Zametki, Vol. 40, No. 4, pp. 527–541, October, 1986.
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Vatutin, V.A. Critical Bellman-Harris branching processes starting with a large number of particles. Mathematical Notes of the Academy of Sciences of the USSR 40, 803–811 (1986). https://doi.org/10.1007/BF01159675
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DOI: https://doi.org/10.1007/BF01159675