Convergence of a sequence of nearly critical branching processes with immigration
Author:
Ya. M. Khusanbaev
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 90 (2015), 201-205
MSC (2010):
Primary 60J80; Secondary 62F12
DOI:
https://doi.org/10.1090/tpms/960
Published electronically:
August 11, 2015
MathSciNet review:
3242031
Full-text PDF Free Access
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Abstract: We study a sequence of nearly critical branching processes with immigration in the case where the rate of convergence of the expectation of the number of offsprings to 1 is slower than $n^{-1}$. We provide sufficient conditions under which these processes converge in probability to a nonrandom process and prove a limit theorem for the fluctuations of nearly critical branching processes.
References
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- S. A. Aliev, A limit theorem for Galton-Watson branching processes with immigration, Ukrain. Mat. Zh. 37 (1985), no. 5, 656–659, 679 (Russian). MR 815316
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- Ya. M. Khusanbaev, On the convergence of Galton-Watson branching processes with immigration to a diffusion process, Teor. Ĭmovīr. Mat. Stat. 79 (2008), 162–167 (Russian, with Russian summary); English transl., Theory Probab. Math. Statist. 79 (2009), 179–185. MR 2494547, DOI https://doi.org/10.1090/S0094-9000-09-00777-7
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References
- K. Kawazu and S. Watanabe, Branching processes with immigration and related limit theorems, Teor. Veroyatnost. Primenen. XVI (1971), no. 2, 34–51; English transl. in Theory Probab. Appl. 16, no. 1, 36–54. MR 0290475 (44:7656)
- S. A. Aliev, A limit theorem for the Galton–Watson branching processes with immigration, Ukrain. Matem. Zh. 37 (1985), 656–659; English transl. in Ukrainian Math. J. 37 (1985), no. 5, 535–538. MR 815316 (87d:60074)
- T. N. Sriram, Invalidity of bootstrap for critical branching process with immigration, Ann. Statist. 22 (1994), 1013–1023. MR 1292554 (95g:62098)
- M. Ispany, G. Pap, and M. C. A. Van Zuijlen, Fluctuation limit of branching processes with immigration and estimation of the means, Adv. Appl. Probab. 37 (2005), 523–528. MR 2144565 (2006c:60103)
- Ya. M. Khusanbaev, On the convergence of Galton–Watson branching processes with immigration to a diffusion process, Teor. Imovir. Matem. Statist. 79 (2008), 183–189; English transl. in Theor. Probability and Math. Statist. 79 (2009), 179–185. MR 2494547 (2010a:60291)
- R. S. Liptser and A. N. Shiryaev, Theory of Martingales, Moscow, “Nauka”, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664 (90j:60046)
- A. A. Borovkov, Theory of Probability, “Nauka”, Moscow, 1986; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998.
- P. Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0233396 (38:1718)
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Additional Information
Ya. M. Khusanbaev
Affiliation:
Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan
Email:
yakubjank@mail.ru
Keywords:
Branching process with immigration,
week convergence
Received by editor(s):
January 7, 2012
Published electronically:
August 11, 2015
Article copyright:
© Copyright 2015
American Mathematical Society
