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Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

Part of: Applications Markov processes Limit theorems

Published online by Cambridge University Press:  22 June 2017

Ollivier Hyrien ,
Kosto V. Mitov  and
Nikolay M. Yanev
Ollivier Hyrien*
Affiliation:
Fred Hutchinson Cancer Research Center
Kosto V. Mitov*
Affiliation:
Vasil Levski National Military University
Nikolay M. Yanev*
Affiliation:
Bulgarian Academy of Sciences
*
* Postal address: Program in Biostatistics, Bioinformatics, and Epidemiology, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave. N., Seattle, WA 98109, USA. Email address: ohyrien@fredhutch.org
** Postal address: Faculty of Aviation, Vasil Levski National Military University, 5856 D. Mitropolia, Pleven, Bulgaria. Email address: kmitov@yahoo.com
*** Postal address: Department of Operations Research, Probability and Statistics, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria. Email address: yanev@math.bas.bg
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Abstract

We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

Keywords

Branching processimmigrationPoisson processlimit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60J85: Applications of branching processes 62P10: Applications to biology and medical sciences
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 569 - 587
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Ahsanullah, M. and Yanev, G. P. (eds) (2008). Records and Branching Processes. Nova Science, New York. Google Scholar
[2] Assmusen, S. and Hering, H. (1983). Branching Processes (Progr. Prob. Statist. 3). Birkhäuser, Boston. CrossRefGoogle Scholar
[3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. Google Scholar
[4] Chen, R., Hyrien, O., Noble, M. and Mayer-Pröschel, M. (2011). A composite likelihood approach to the analysis of longitudinal clonal data on multitype cellular systems under an age-dependent branching process. Biostatist. 12, 173–191. Google Scholar
[5] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York. Google Scholar
[6] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin. Google Scholar
[7] Hyrien, O. and Yanev, N. M. (2010). Modeling cell kinetics using branching processes with nonhomogeneous Poisson immigration. C. R. Acad. Bulg. Sci. 63, 1405–1414. Google Scholar
[8] Hyrien, O. and Yanev, N. M. (2015). Age-dependent branching processes with non-homogeneous Poisson immigration as models of cell kinetics. In Modeling and Inference in Biomedical Sciences: In Memory of Andrei Yakovlev, eds D. Oakes, W. J. Hall and A. Almudevar, Institute of Mathematical Statistics, Beachwood, OH. Google Scholar
[9] Hyrien, O., Dietrich, J. and Noble, M. (2010). Quantitative analysis of the effect of chemotherapy on division and differentiation of glial precursor cells. Cancer Res. 70, 10051–10059. Google Scholar
[10] Hyrien, O., Mitov, K. M. and Yanev, N. M. (2015). CLT for Sevastyanov branching processes with non-homogeneous immigration. J. Appl. Statist. Sci. 21, 229–237. Google Scholar
[11] Hyrien, O., Mitov, K. M. and Yanev, N. M. (2016). Supercritical Sevastyanov branching processes with non-homogeneous Poisson immigration. In Branching Processes and their Applications (Lecture Notes Statist. 219), eds I. M. del Puerto et al., Springer, New York, pp. 151–166. Google Scholar
[12] Hyrien, O., Yanev, N. M. and Jordan, C. T. (2015). A test of homogeneity for age-dependent branching processes with immigration. Electron. J. Statist. 9, 898–925. CrossRefGoogle ScholarPubMed
[13] Hyrien, O., Chen, R., Mayer-Pröschel, M. and Noble, M. (2010). Saddlepoint approximating to the moments of multitype age-dependent branching processes, with applications. Biometrics 66, 567–577. CrossRefGoogle Scholar
[14] Hyrien, O., Mayer-Pröschel, M., Noble, M. and Yakovlev, A. (2005). A stochastic model to analyze clonal data on multi-type cell populations. Biometrics 61, 199–207. Google Scholar
[15] Hyrien, O., Peslak, S. A., Yanev, N. M. and Palis, J. (2015). Stochastic modeling of stress erythropoiesis using a two-type age-dependent branching process with immigration. J. Math. Biol. 70, 1485–1521. Google Scholar
[16] Hyrien, O. et al. (2006). Stochastic modeling of oligodendrocyte generation in cell culture: model validation with time-lapse data. Theoret. Biol. Medical Modelling 3, 21pp. Google Scholar
[17] Jagers, P. (1968). Age-dependent branching processes allowing immigration. Theory Prob. Appl. 13, 225–236. Google Scholar
[18] Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London. Google Scholar
[19] Kaplan, N. and Pakes, A. G. (1974). Supercritical age-dependent branching processes with immigration. Stoch. Process. Appl. 2, 371–389. Google Scholar
[20] Kesten, H. and Stigum, B. P. (1967). Limit theorems for decomposable multi-dimensional Galton–Watson processes. J. Math. Anal. Appl. 17, 309–338. Google Scholar
[21] Mitov, K. V. and Yanev, N. M. (1985). Bellman–Harris branching processes with state-dependent immigration. J. Appl. Prob. 22, 757–765. CrossRefGoogle Scholar
[22] Mitov, K. V. and Yanev, N. M. (1989). Bellman–Harris branching processes with a special type of state-dependent immigration. Adv. Appl. Prob. 21, 270–283. Google Scholar
[23] Mitov, K. V. and Yanev, N. M. (2002). Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration. Stoch. Models 18, 281–300. CrossRefGoogle Scholar
[24] Mitov, K. V. and Yanev, N. M. (2013). Sevastyanov branching processes with non-homogeneous Poisson immigration. Proc. Steklov Inst. Math. 282, 172–185. Google Scholar
[25] Olofsson, P. (1996). General branching processes with immigration. J. Appl. Prob. 33, 940–948. Google Scholar
[26] Pakes, A. G. (1972). Limit theorems for an age-dependent branching process with immigration. Math. Biosci. 14, 221–234. Google Scholar
[27] Pakes, A. G. and Kaplan, N. (1974). On the subcritical Bellman–Harris process with immigration. J. Appl. Prob. 11, 652–668. CrossRefGoogle Scholar
[28] Radcliffe, J. (1972). The convergence of a super-critical age-dependent branching process allowing immigration at the epochs of a renewal process. Math. Biosci. 14, 37–44. Google Scholar
[29] Sevast'yanov, B. A. (1957). Limit theorems for branching random processes of special form. Teor. Veroyat. Primen. 2, 339–348 (in Russian). English translation: Theory Prob. Appl. 2, 321–331. Google Scholar
[30] Sevast'yanov, B. A. (1964). Age-dependent branching processes. Theory Prob. Appl. 9, 521–537. (Addendum: 11 (1966), 321–322. Google Scholar
[31] Sevast'yanov, B. A. (1968). Limit theorems for age-dependent branching processes. Theory Prob. Appl. 13, 237–259. CrossRefGoogle Scholar
[32] Sevast'yanov, B. A. (1971). Branching Processes. Nauka, Moscow (in Russian). Google Scholar
[33] Yakovlev, A. and Yanev, N. (2006). Branching stochastic processes with immigration in analysis of renewing cell populations. Math. Biosci. 203, 37–63. Google Scholar
[34] Yakovlev, A. and Yanev, N. (2007). Age and residual lifetime distributions for branching processes. Statist. Prob. Lett. 77, 503–513. Google Scholar
[35] Yanev, N. M. (1971). Branching processes allowing immigration. Bull. Inst. Math. Acad. Bulg. Sci. 15, 71–88 (in Bulgarian). Google Scholar
[36] Yanev, N. M. (1972). On a class of decomposable age-dependent branching processes. Math. Balkanica 2, 58–75 (in Russian). Google Scholar