Some limit theorems for controlled branching processes
Author:
Ya. M. Khusanbaev
Translated by:
O. Klesov
Journal:
Theor. Probability and Math. Statist. 81 (2010), 51-58
MSC (2000):
Primary 60J80; Secondary 60F17, 60J65
DOI:
https://doi.org/10.1090/S0094-9000-2010-00809-X
Published electronically:
January 18, 2011
MathSciNet review:
2667309
Full-text PDF Free Access
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Additional Information
Abstract: Limit theorems are obtained for branching processes depending on the size of the population. These results contain the case of convergence of such processes to the deterministic limit. Functional central limit theorems for fluctuations are proved.
References
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References
- P. Haccou, P. Jagers, and V. A. Vatutin, Branching Processes. Variation, Growth, and Extinction of Populations, Cambridge University Press, 2005. MR 2429372 (2009h:92064)
- Joshua B. Levy, Transience and recurrence of state-dependent branching processes with an immigration component, Adv. Appl. Probab. 11 (1979), 73–92. MR 517552 (80b:60111a)
- G. Kersting, On recurrence and transience of growth models, J. Appl. Probab. 23 (1986), 614–625. MR 855369 (88h:60086)
- G. Kersting, Some properties of stochastic difference equations, Stochastic Modeling in Biology (P. Tautu, ed.), World Scientific, Singapore, 1990, pp. 328–339. MR 1090847 (92c:60091)
- P. Küster, Asymptotic growth of controlled Galton–Watson processes, Ann. Probab. 13 (1985), no. 4, 1157–1178. MR 806215 (87c:60072)
- F. C. Klebaner, A limit theorem for population-size-dependent branching processes, J. Appl. Probab. 22 (1985), 48–57. MR 776887 (86h:60166)
- F. C. Klebaner, Geometric rate of growth in population-size-dependent branching processes, J. Appl. Probab. 21 (1984), 40–49. MR 732669 (85f:60124)
- D. Pierre-Loti-Viaud, A strong law and a central limit theorem for controlled Galton–Watson processes, J. Appl. Probab. 31 (1994), 22–37. MR 1260568 (94m:60175)
- N. Lalam and C. Jacob, Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process, Adv. Appl. Probab. 36 (2004), 582–601. MR 2058150 (2005f:60182)
- N. Lalam, C. Jacob, and P. Jagers, Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency, Adv. Appl. Probab. 36 (2004), 602–615. MR 2058151 (2004m:60184)
- A. N. Shiryaev, Probability, Nauka, Moscow, 1980; English transl., Springer-Verlag, New York, 1996. MR 1368405 (97c:60003)
- R. Sh. Liptser and A. N. Shiryaev, Theory of Martingales, Nauka, Moscow, 1986; English transl., Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1022664 (90j:60046)
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0233396 (38:1718)
- B. A. Sevast’yanov and A. M. Zubkov, Controlled branching processes, Teor. Veroyatnost. i Primenen. 19 (1974), no. 1, 15–25; English transl. in Theory Probab. Appl. 19 (1974), no. 1, 11–24. MR 0339350 (49:4109)
- A. M. Zubkov, Analogies between Galton–Watson processes and $\varphi$-branching processes, Teor. Veroyatnost. i Primenen. 19 (1974), no. 2, 319–339; English transl. in Theory Probab. Appl. 19 (1974), no. 2, 309–331. MR 0353480 (50:5963)
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Additional Information
Ya. M. Khusanbaev
Affiliation:
Department of Probability Theory and Mathematical Statistics, Institute of Mathematics and Information Technologies, Durmon Yuli Street 29, Tashkent 100125, Uzbekistan
Email:
yakubjank@mail.ru
Keywords:
Population-size-dependent branching process,
weak convergence,
Wiener process,
limit theorem
Received by editor(s):
August 6, 2007
Published electronically:
January 18, 2011
Article copyright:
© Copyright 2010
American Mathematical Society
