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 Tr. Mat. Inst. Steklova, 2015, Volume 290, Pages 114–135 (Mi tm3632)

Decomposable branching processes with a fixed extinction moment

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: The asymptotic behavior as $n\to \infty$ of the probability of the event that a decomposable critical branching process $\mathbf Z(m)= (Z_1(m),…,Z_N(m))$, $m=0,1,2,…$, with $N$ types of particles dies at moment $n$ is investigated, and conditional limit theorems are proved that describe the distribution of the number of particles in the process $\mathbf Z(\cdot )$ at moment $m<n$ given that the extinction moment of the process is $n$. These limit theorems can be considered as statements describing the distribution of the number of vertices in the layers of certain classes of simply generated random trees of fixed height.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968515030103

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 290:1, 103–124

Bibliographic databases:

Document Type: Article
UDC: 519.218.24

Citation: V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with a fixed extinction moment”, Modern problems of mathematics, mechanics, and mathematical physics, Collected papers, Tr. Mat. Inst. Steklova, 290, MAIK Nauka/Interperiodica, Moscow, 2015, 114–135; Proc. Steklov Inst. Math., 290:1 (2015), 103–124

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3632
• https://doi.org/10.1134/S0371968515030103
• http://mi.mathnet.ru/eng/tm/v290/p114

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1. Vladimir A. Vatutin, Elena E. Dyakonova, “Extinction of decomposable branching processes”, Discrete Math. Appl., 26:3 (2016), 183–192
2. V. I. Afanasyev, “On a decomposable branching process with two types of particles”, Proc. Steklov Inst. Math., 294 (2016), 1–12
3. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
4. C. Smadi, V. A. Vatutin, “Reduced two-type decomposable critical branching processes with possibly infinite variance”, Markov Process. Relat. Fields, 22:2 (2016), 311–358
5. V. A. Vatutin, “A Conditional Functional Limit Theorem for Decomposable Branching Processes with Two Types of Particles”, Math. Notes, 101:5 (2017), 778–789
6. G. K. Kobanenko, “Predelnye teoremy dlya ogranichennykh vetvyaschikhsya protsessov”, Diskret. matem., 29:2 (2017), 18–28
7. V. A. Vatutin, E. E. D'yakonova, “Decomposable branching processes with two types of particles”, Discrete Math. Appl., 28:2 (2018), 119–130
8. E. E. D'yakonova, “A subcritical decomposable branching process in a mixed environment”, Discrete Math. Appl., 28:5 (2018), 275–283
9. V. I. Afanasyev, “A Functional Limit Theorem for Decomposable Branching Processes with Two Particle Types”, Math. Notes, 103:3 (2018), 337–347
10. M. Liu, V. A. Vatutin, “Reduced critical branching processes for small populations”, Theory Probab. Appl., 63:4 (2019), 648–656
11. V. A. Vatutin, “Uslovnaya predelnaya teorema dlya blizkikh k kriticheskim vetvyaschikhsya protsessov s finalnym tipom chastits”, Matem. vopr. kriptogr., 9:4 (2018), 53–72
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