On asymptotics of branching processes with immigration
-
Yakubdzhan M. Khusanbaev
Abstract
We consider a sequence of almost critical branching processes with immigration supposing that the immigration process is weakly stationary. The rate of growth and asymptotic properties of fluctuations of such branching processes are investigated.
Originally published in Diskretnaya Matematika (2016) 28, №1, 113–122 (in Russian).
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 16-01-00226-a
Funding statement: This work was supported by the Russian Foundation for Basic Research, grant nos. 16-01-00226-a and 16-01-00470-a).
References
[1] Athreya K.B., Ney P.E., Branching Processes, Springer-Verlag, New York, 1972, 288 pp.10.1007/978-3-642-65371-1Search in Google Scholar
[2] Seneta E., “An explicit-limit theorem for the critical Galton-Watson process with Immigration”, J. Roy. Stat. Soc., 32 (1970), 149–152.10.1111/j.2517-6161.1970.tb00826.xSearch in Google Scholar
[3] Wei C.Z., Winicki J., “Some asymptotic results for the branching process with immigration”, Stoch. Process. Appl., 31 (1989), 261–282.10.1016/0304-4149(89)90092-6Search in Google Scholar
[4] Rakhimov I., “Functional limit theorems for critical processes with immigration”, Adv. Appl. Probab., 39 (2007), 1054–1069.10.1239/aap/1198177239Search in Google Scholar
[5] Khusanbaev Ya. M., “On the asymptotical behavior of a subcritical branching process with immigration”, Ukr. Math. J., 65:6 (2013), 928–937.10.1007/s11253-013-0829-zSearch in Google Scholar
[6] Nagaev S. V., “A limit theorem for branching processes with immigration”, Theory Probab. Appl., 20:1 (1975), 176–179.10.1137/1120019Search in Google Scholar
[7] Sriram T.N., “Invalidity of bootstrap for critical branching process with immigration”, Ann. Statist., 22 (1994), 1013–1023.10.1214/aos/1176325509Search in Google Scholar
[8] Ispany M., Pap G. and Van Zuijlen M.C.A., “Fluctuation limit of branching processes with immigration and estimation of the mean”, Adv. Appl. Probab., 37 (2005), 523–528.10.1239/aap/1118858637Search in Google Scholar
[9] Khusanbaev Ya. M., “Almost critical branching processes and limit theorems”, Ukr. Math. J., 61:1 (2009), 154–163.10.1007/s11253-009-0191-3Search in Google Scholar
[10] Billingsley P., Convergence of probability measures, 2, Wiley, 1968, 253 pp.Search in Google Scholar
[11] Shiryaev A. N., “Probability”, Graduate Texts in Mathematics, 95, 2, Springer-Verlag, New York, 1996, 624 pp.10.1007/978-1-4757-2539-1Search in Google Scholar
[12] Sirajdinov S.Kh., Formanov Sh.K., Limit theorems for sums of random vectors linked into aMarkov chain, Tashkent: FAN, 1979 (in Russian), 171 pp.Search in Google Scholar
[13] Khusanbaev Ya. M., “On the convergence rate in one limit theorem for branching processes with immigration”, Siberian Math. J., 55:1 (2014), 178–184.10.1134/S0037446614010200Search in Google Scholar
[14] Asadullin M. Kh., Nagaev S. V., “Limit theorems for a critical branching process with immigration”,Math. Notes, 32:4 (1982), 750–757.10.1007/BF01152384Search in Google Scholar
[15] Badalbaev I. S., Zubkov A. M., “Limit theorems for a sequence of branching processes with immigration”, Theory Probab. Appl., 28:2 (1984), 404–409.10.1137/1128034Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Extension of the Rissanen algorithm to the factorization of block-Hankel matrices for solving systems of linear equations
- On asymptotics of branching processes with immigration
- On some measures of complexity of finite Abelian groups
- Complexity classification of the edge coloring problem for a family of graph classes
- On FE-precomplete classes in countable-valued logic
- On 1-stable perfectly balanced Boolean functions
- On bases of closed classes of vector functions of many-valued logic
- On the asymptotic normality of some sums of dependent random variables
- On serial rings
Articles in the same Issue
- Frontmatter
- Extension of the Rissanen algorithm to the factorization of block-Hankel matrices for solving systems of linear equations
- On asymptotics of branching processes with immigration
- On some measures of complexity of finite Abelian groups
- Complexity classification of the edge coloring problem for a family of graph classes
- On FE-precomplete classes in countable-valued logic
- On 1-stable perfectly balanced Boolean functions
- On bases of closed classes of vector functions of many-valued logic
- On the asymptotic normality of some sums of dependent random variables
- On serial rings
