The convergence of Galton–Watson branching processes with immigration to a diffusion process
Author:
Ya. M. Khusanbaev
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 79 (2009), 179-185
MSC (2000):
Primary 60J80; Secondary 60F17, 60J60
DOI:
https://doi.org/10.1090/S0094-9000-09-00777-7
Published electronically:
December 30, 2009
MathSciNet review:
2494547
Full-text PDF Free Access
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Additional Information
Abstract: A sequence of almost critical Galton–Watson branching processes with immigration is studied. Sufficient conditions for the weak convergence of such processes to a diffusion process are found.
References
- 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
- 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
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1981. MR 637061
References
- C. Z. Wei and J. Winnicki, Some asymptotic results for the branching process with immigration, Stoch. Process. Appl. 31 (1989), 261–282. MR 998117 (91c:60109)
- 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), 523–538. MR 2144565 (2006c:60103)
- S. N. Ethier and T. G. Kurtz, Markov Processes, John Wiley, New York, 1986. MR 838085 (88a:60130)
- N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Kodansha, Ltd., Amsterdam–New York–Tokyo, 1981. MR 637061 (84b:60080)
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Additional Information
Ya. M. Khusanbaev
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute of Mathematics and Information Technologies, F. Khodzhaev Street 29, Tashkent 100125, Uzbekistan
Email:
yakubjank@mail.ru
Keywords:
Almost critical Galton–Watson branching process with immigration,
weak convergence,
stochastic differential equation,
Skorokhod’s space
Received by editor(s):
August 6, 2007
Published electronically:
December 30, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
