On the asymptotic behavior of branching processes with stationary immigration

Authors

Yakubjan Khusanbaev, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of SciencesFollow
Sadulla Sharipov, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of SciencesFollow

Abstract

In this paper, we consider nearly critical branching processes with immigration. We study the convergence of a sequence of nearly critical branching processes with immigration when immigration is a stationary in wide sense. Moreover, we derive an asymptotic for characteristic function of this process.

First Page

59

Last Page

73

References

1. Athreya K.B., Vidyashankar A.N. Branching process. Handbook of statistics. Vol. 19, 35–53 (2001).

2. Athreya K.B., Jagers P. Classical and modern branching processes. IMA volumes in Mathematics and Applications, Springer–Verlag, New York (1997).

3. Athreya K.B., Jagers P. Classical and modern branching processes. IMA volumes in Mathematics and Applications, Springer–Verlag, New York (1997).

4. Haccou P., Jagers P., Vatutin V. Branching processes. Variation. growth and extinction of population, Cambridge University Press, Cambridge (2005).

5. Vatutin V., Zubkov A. Branching processes. I. Journal of Soviet Mathematics. Vol. 31, Issue 1, 2431–2474 (1987).

6. Sevast'yanov B.A. Limit theorems for branching processes of special form. Theory of Probability and its Applications. Vol. 2, Issue 3, 321–331 (1957).

7. Seneta E. An explicit–limit theorem for the critical Galton–Watson process with immigration. Journal of the Royal Statistical Society. Series B (Methodological). Vol. 32,Issue 1, 149–152 (1970).

8. Pakes A.G. On the critical Galton–Watson process with immigration. Journal Australian Mathematical Society. Vol. 12, Issue 4, 476–482 (1971).

9. Pakes A.G. Some new limit theorems for the critical branching process allowing immigration. Stochastic processes and their Applications. Vol. 4, Issue 2, 175–185 (1976).

10. Vatutin V. A conditional limit theorem for a critical branching process with immigration. Mathematical Notes of the Academy of Sciences of the USSR. Vol. 21, Issue 5, 405–411 (1977).

11. Pakes A.G. Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean. Advances in Applied Probability. Vol. 11, Issue 1, 31–62 (1979).

12. Nagaev S.V. A limit theorem for branching processes with immigration. Theory of Probability and its Applications. Vol. 20, Issue 1, 176–179 (1975).

13. Asadullin M.Kh., Nagaev S.V. Limit theorems for a critical branching process with immigration. Mathematical notes of the Academy of Sciences of the USSR. Vol. 32,Issue 4, 750–757 (1975).

14. Badalbaev I.S., Zubkov A. Limit Theorems for a Sequence of Branching Processes with Immigration. Theory of Probability and its Applications. Vol. 28, Issue 2, 404–409 (1983).

15. Feller W. Diffusion processes in genetics. Proceedings Second Berkeley Symp Math. Stat. Prob., University of California Press, 227–246 (1951).

16. Lindvall T. Limit theorems for some functionals of certain Galton–Watson branching processes. Advances of Applied Probability. Vol. 6, Issue 2, 309–321 (1974).

17. Kawazu K., Watanabe Sh. Branching Processes with Immigration and Related Limit Theorems. Theory of Probability and its Applications. Vol. 16, Issue 1, 36–54 (1971).

18. Li Z.H. Branching Processes with Immigration and Related Topics. Frontiers of Mathematics in China. Vol. 1, Issue 1, 73–97 (2006).

19. Grimvall A. On the Convergence of Sequences of Branching Processes. The Annals of Probability. Vol. 2, Issue 6, 1027–1045 (1974).

20. Sriram T.N. Invalidity of Bootstrap for critical branching processes with immigration. The Annals of Statistics. Vol. 22, Issue 2, 1013–1023. (1994).

21. Wei C.Z., Winnicki J. A unified estimation theory for the branching process with immigration. Technical Repport, Univ. Maryland. (1987).

22. Wei C.Z., Winnicki J. Some asymptotic results for the branching process with immigration. Stochastic processes and their Applications. Vol. 31, Issue 2, 261–282 (1989).

23. Li Z.H. Ornstein–Uhlenbeck type processes and branching processes with immigration. Journal of Applied Probability. Vol. 37, Issue 3, 627–634 (2000).

24. Ispany M., Pap G., Van Zuijlen M.C.A. Fluctuation limit theorem of branching processes with immigration and estimation of the mean. Adv.Appl.Probab. Vol. 37, Issue 2, 523–538 (2005).

25. Li Y.Q. A fluctuation type limit theorem for Jirina processes with immigration. Acta Mathematica Sinica. Vol. 25, Issue 8, 1379–1388 (2009).

26. Chan N.H., Wei Ch.Z. Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics. Vol. 22, Issue 15, 1050–1063 (1987).

27. Khusanbaev Y.M. On the convergence rate in one limit theorem for branching processes with immigration. Siberian Mathematical Journal, Vol. 55, Issue 1, 178–184 (2014).

28. Khusanbaev Y.M. On asymptotics of branching processes with immigration. Discrete Mathematics and Applications. Vol. 28, Issue 1, 113–122 (2016).

29. Shiryaev A.N. Probability, Graduate Texts in Mathematics. Springer–Verlag, New York (1996).

30. Billingsley P. Convergence of Probability Measures. Wiley Series in Probability and Mathematical Statistics, Wiley, New York (1968).

Recommended Citation

Khusanbaev, Yakubjan and Sharipov, Sadulla (2020) "On the asymptotic behavior of branching processes with stationary immigration," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3: Iss. 1, Article 6.
DOI: https://doi.org/10.56017/2181-1318.1054