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Branching processes in a random environment with immigration stopped at zero

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  04 May 2020

Elena Dyakonova ,
Doudou Li ,
Vladimir Vatutin  and
Mei Zhang
Elena Dyakonova*
Affiliation:
Steklov Mathematical Institute
Doudou Li*
Affiliation:
Beijing Normal University
Vladimir Vatutin*
Affiliation:
Steklov Mathematical Institute and Beijing Normal University
Mei Zhang*
Affiliation:
Beijing Normal University
*
*Postal address: Steklov Mathematical Institute, 8 Gubkin St., Moscow, 119991, Russia.
***Postal address: School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, P.R. China.
*Postal address: Steklov Mathematical Institute, 8 Gubkin St., Moscow, 119991, Russia.
**Email address: [email protected]
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Abstract

A critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, i.e. the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time.

Keywords

Branching processrandom environmentimmigrationlife period

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 237 - 249
Copyright
© Applied Probability Trust 2020

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References

Afanasyev, V. I. (2014). Conditional limit theorem for maximum of random walk in a random environment. Theory Prob. Appl. 58, 525–545.CrossRefGoogle Scholar
Afanasyev, V. I., Boeinghoff, Ch., Kersting, G. and Vatutin, V. A. (2012). Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Prob. 25, 703–732.10.1007/s10959-010-0331-6CrossRefGoogle Scholar
Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in random environment. Ann. Prob. 33, 645–673.CrossRefGoogle Scholar
Badalbaev, I. S. and Mashrabbaev, A. (1983). Lifetimes of an $r \gt 1$-type Galton–Watson process with immigration. Izv. Akad. Nauk UzSSR, Ser. Fiz., Mat. Nauk. 2, 7–13.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
Haccou, P., Jagers, P. and Vatutin, V. A. (2005) Branching Processes in Biology: Evolution, Growth and Extinction (Camb. Ser. Adaptive Dynamics), 5. Cambridge University Press.CrossRefGoogle Scholar
Kagan, Y. Y. (2010) Statistical distributions of earthquake numbers: consequence of branching process. Geophys. J. Int., 180, 1313–1328.CrossRefGoogle Scholar
Kaplan, N. (1974). Some results about multidimensional branching processes with random environments. Ann. Prob. 2, 441–455.CrossRefGoogle Scholar
Kersting, G. and Vatutin, V. (2017). Discrete Time Branching Processes in Random Environment, ISTE & Wiley, New York.CrossRefGoogle Scholar
Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit law for random walk in a random environment. Comp. Math. 30, 145–168.Google Scholar
Key, E. S. (1987). Limiting distributions and regeneration times for multitype branching processes with immigration in a random environment. Ann. Prob. 15, 344–353.10.1214/aop/1176992273CrossRefGoogle Scholar
Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology. Springer, New York.CrossRefGoogle Scholar
Mitov, K. V. (1982). Conditional limit theorem for subcritical branching processes with immigration. In Matem. i Matem. Obrazov. Dokl. ii Prolet. Konf. C”yuza Matem. B”lgarii, Sl”nchev Bryag, 6–9 Apr. Sofia, pp. 398–403.Google Scholar
Novick, R. P. and Hoppenstead, F. C. (1978) On plasmid incompatibility. Plasmid 1, 431–434.CrossRefGoogle ScholarPubMed
Rogozin, B. A. (1962). The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Prob. Appl. 16, 575–595.CrossRefGoogle Scholar
Roitershtein, A. (2007) A note on multitype branching processes with immigration in a random environment. Ann. Prob. 35, 1573–1592.CrossRefGoogle Scholar
Seneta, E. and Tavare, S. (1983). A note on models using the branching process with immigration stopped at zero. J. Appl. Prob. 20, 11–18.CrossRefGoogle Scholar
Tanny, D. (1981). On multitype branching processes in a random environment. Adv. Appl. Prob. 13, 464–497.10.2307/1426781CrossRefGoogle Scholar
Vatutin, V. A. (1977). A conditional limit theorem for a critical branching process with immigration. Math. Notes 21, 405–411.CrossRefGoogle Scholar
Zolotarev, V. M. (1957). Mellin–Stiltjes transform in probability theory. Theory Prob. Appl. 2, 433–460.CrossRefGoogle Scholar
Zubkov, A. M. (1972). Life-periods of a branching process with immigration. Theory Prob. Appl. 17, 174–183.10.1137/1117018CrossRefGoogle Scholar