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
- Patsy Haccou, Peter Jagers, and Vladimir A. Vatutin, Branching processes: variation, growth, and extinction of populations, Cambridge Studies in Adaptive Dynamics, vol. 5, Cambridge University Press, Cambridge; IIASA, Laxenburg, 2007. MR 2429372
- Joshua B. Levy, Transience and recurrence of state-dependent branching processes with an immigration component, Adv. in Appl. Probab. 11 (1979), no. 1, 73–92. MR 517552, DOI https://doi.org/10.2307/1426769
- G. Kersting, On recurrence and transience of growth models, J. Appl. Probab. 23 (1986), no. 3, 614–625. MR 855369, DOI https://doi.org/10.2307/3214001
- Götz Kersting, Some properties of stochastic difference equations, Stochastic modelling in biology (Heidelberg, 1988) World Sci. Publ., Teaneck, NJ, 1990, pp. 328–339. MR 1090847
- Petra Küster, Asymptotic growth of controlled Galton-Watson processes, Ann. Probab. 13 (1985), no. 4, 1157–1178. MR 806215
- F. C. Klebaner, A limit theorem for population-size-dependent branching processes, J. Appl. Probab. 22 (1985), no. 1, 48–57. MR 776887, DOI https://doi.org/10.2307/3213747
- F. C. Klebaner, Geometric rate of growth in population-size-dependent branching processes, J. Appl. Probab. 21 (1984), no. 1, 40–49. MR 732669, DOI https://doi.org/10.2307/3213662
- Daniel Pierre-Loti-Viaud, A strong law and a central limit theorem for controlled Galton-Watson processes, J. Appl. Probab. 31 (1994), no. 1, 22–37. MR 1260568, DOI https://doi.org/10.1017/s0021900200107302
- N. Lalam and C. Jacob, Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process, Adv. in Appl. Probab. 36 (2004), no. 2, 582–601. MR 2058150, DOI https://doi.org/10.1239/aap/1086957586
- N. Lalam, C. Jacob, and P. Jagers, Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency, Adv. in Appl. Probab. 36 (2004), no. 2, 602–615. MR 2058151, DOI https://doi.org/10.1239/aap/1086957587
- A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405
- 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
- Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
- B. A. Sevast′janov and A. M. Zubkov, Controlled branching processes, Teor. Verojatnost. i Primenen. 19 (1974), 15–25 (Russian, with English summary). MR 0339350
- A. M. Zubkov, Analogies between Galton-Watson processes and $\varphi $-branching processes, Teor. Verojatnost. i Primenen. 19 (1974), 319–339 (Russian, with English summary). MR 0353480
- F. C. Klebaner, On population-size-dependent branching processes, Adv. in Appl. Probab. 16 (1984), no. 1, 30–55. MR 732129, DOI https://doi.org/10.2307/1427223
- A. A. Borovkov, Teoriya veroyatnosteĭ, 2nd ed., “Nauka”, Moscow, 1986 (Russian). MR 891184
- Tetsuo Fujimagari, Controlled Galton-Watson process and its asymptotic behavior, K\B{o}dai Math. Sem. Rep. 27 (1976), no. 1-2, 11–18. MR 400435
- L. V. Levina, A. M. Leontovich, and I. I. Pyatetskii-Shapiro, A controllable branching process, Problems Inform. Transmission 4 (1968), no. 2, 55–64 (1971). MR 283891, DOI https://doi.org/10.1016/s0016-0032%2815%2991347-9
- V. A. Labkovskiĭ, A limit theorem for generalized branching random processes that depend on population size, Teor. Verojatnost. i Primenen. 17 (1972), 71–83 (Russian, with English summary). MR 0298785
- F. C. Klebaner, Population-size-dependent branching process with linear rate of growth, J. Appl. Probab. 20 (1983), no. 2, 242–250. MR 698528, DOI https://doi.org/10.2307/3213798
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)
- F. C. Klebaner, On population-size-dependent branching processes, Adv. Appl. Probab. 16 (1984), 30–55. MR 732129 (85g:60090)
- A. A. Borovkov, Probability theory, Moscow, Nauka, 1986; English transl., Gordon and Breach Science Publishers, Amsterdam, 1998. MR 891184 (88c:60001)
- T. Fujimagari, Controlled Galton–Watson process and its asymptotic behavior, Kodai Math. Sem. Rep. 27 (1976), 11–18. MR 0400435 (53:4268)
- L. V. Levina, A. M. Leontovich, and I. I. Pyatetskiĭ-Shapiro, A controllable branching process, Problemy Peredachi Informacii IV (1968), no. 2, 72–82; English transl. in Problems of Information Transmission IV (1971), no. 2, 55–64. MR 0283891 (44:1121)
- V. A. Labkovskiĭ, A limit theorem for generalized random branching processes dependent on the size of the population, Teor. Veroyatnost. i Primenen. 17 (1972), no. 1, 71–83; English transl. in Theory Probab. Appl. 17 (1972), no. 1, 72–85. MR 0298785 (45:7834)
- F. C. Klebaner, Population-size-dependent branching process with linear rate of growth, J. Appl. Probab. 20 (1983), 242–250. MR 698528 (84g:60139)
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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