The asymptotic behavior of the total number of particles in a critical branching process with immigration
Author:
Ya. M. Khusanbaev
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 96 (2018), 169-176
MSC (2010):
Primary 60J80; Secondary 60F17
DOI:
https://doi.org/10.1090/tpms/1042
Published electronically:
October 5, 2018
MathSciNet review:
3666880
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Abstract: A sequence of branching processes with immigration is considered in the case where the mean number of descendents of a particle tends to unity. The rate of growth and asymptotic behavior of the total number of particles in the population are found.
References
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- T. N. Sriram, Invalidity of bootstrap for critical branching processes with immigration, Ann. Statist. 22 (1994), no. 2, 1013–1023. MR 1292554, DOI https://doi.org/10.1214/aos/1176325509
- M. Ispány, G. Pap, and M. C. A. van Zuijlen, Fluctuation limit of branching processes with immigration and estimation of the means, Adv. in Appl. Probab. 37 (2005), no. 2, 523–538. MR 2144565, DOI https://doi.org/10.1239/aap/1118858637
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664
- D. S. Sil′vestrov, Predel′nye teoremy dlya slozhnykh sluchaĭ nykh funktsiĭ, Izdat. ObЪed. “Višča Škola” pri Kiev. Gosudarstv. Univ., Kiev, 1974 (Russian). MR 0415731
- C. Z. Wei and J. Winnicki, Some asymptotic results for the branching process with immigration, Stochastic Process. Appl. 31 (1989), no. 2, 261–282. MR 998117, DOI https://doi.org/10.1016/0304-4149%2889%2990092-6
- Ya. M. Khusanbaev, On the rate of convergence in a limit theorem for branching processes with immigration, Sibirsk. Mat. Zh. 55 (2014), no. 1, 210–227 (Russian, with Russian summary); English transl., Sib. Math. J. 55 (2014), no. 1, 178–184. MR 3220599, DOI https://doi.org/10.1134/s0037446614010200
References
- A. G. Pakes, Some limit theorems for the total progeny of a branching process, Adv. Appl. Probab. 3 (1971), no. 1, 176–192. MR 0283892
- A. V. Karpenko and S. V. Nagaev, Limit theorems for the complete number of descendants in a Galton–Watson branching process, Teor. Veroyatnost. Primenen. 38 (1993), no. 3, 503–528; English transl. in Theory Probab. Appl. 38 (1993), no. 3, 433–455; MR 1404661
- T. N. Sriram, Invalidity of bootstrap for critical branching process with immigration, Ann. Statist. 22 (1994), 1013–1023. MR 1292554
- M. Ispany, G. Pap, and M. C. A. Van Zuijlen, Fluctuation limit of branching processes with immigration and estimation of the mean, Adv. Appl. Probab. 37 (2005), 523–528. MR 2144565
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0233396
- R. Sh. Liptser and A. N. Shiryayev, Theory of Martingales, “Nauka”, Moscow, 1986; English transl. Kluwer Academic Publishers Group, Mathematics and its Applications (Soviet Series), vol. 49, Dordrecht, 1989. MR 1022664
- D. S. Silvestrov, Limit Theorems for Composite Random Functions, “Vyshcha Shkola”, Kiev, 1974. (Russian) MR 0415731
- C. Z. Wei and J. Winicki, Some asymptotic results for the branching process with immigration, Stochastic Process. Appl. 31 (1989), 261–282. MR 998117
- Ya. M. Khusanbaev, On the rate of convergence in a limit theorem for branching processes with immigration, Sibirsk. Matem. Zh. 55 (2014), no. 1, 221–227. English transl. in Sib. Math. J. 55 (2014), no. 1, 178–184. MR 3220599
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Additional Information
Ya. M. Khusanbaev
Affiliation:
V. I. Romanovskiy Institute of Mathematics, 81, Mirzo Ulugbek street, Tashkent, 100041, Uzbekistan
Email:
yakubjank@mail.ru
Keywords:
Branching processes with immigration,
total number of particles,
weak convergence
Received by editor(s):
January 16, 2017
Published electronically:
October 5, 2018
Article copyright:
© Copyright 2018
American Mathematical Society
