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Extinction of decomposable branching processes

  • Vladimir A. Vatutin EMAIL logo and Elena E. Dyakonova
Published/Copyright: July 15, 2016
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Discrete Mathematics and Applications
From the journal Discrete Mathematics and Applications Volume 26 Issue 3

Abstract

The asymptotic behavior, as n → ∞, of the conditional distribution of the number of particles in a decomposable critical branching process Z(m) = (Z1 (m), …, ZN(m)) with N types of particles at moment m = n − k, k = 0(n), is investigated given that the extinction moment of the process equals to n.

Keywords: decomposable branching processes; criticality; conditional limit theorems

Note: Originally published in Diskretnaya Matematika (2015) 27, No_4, 26–37 (in Russian).

Funding source: Russian Science Foundation

Award Identifier / Grant number: 14-50-00005

Funding statement: This work was supported by the Russian Science Foundation under grant no. 14-50-00005

References

[1] Athreya K.B., Ney P.E., Branching Processes, Berlin: Springer, 1972.10.1007/978-3-642-65371-1Search in Google Scholar

[2] Kozlov M.V., “On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment”, Theory Probab. Appl., 21:4 (1977), 791–804.10.1137/1121091Search in Google Scholar

[3] Afanasyev V.I., “Functional limit theorems for the decomposable branching process with two types of particles”, Discrete Math. Appl., 26 (2016), 71–88.10.1515/dma-2016-0006Search in Google Scholar

[4] Vatutin V.A., Dyakonova E.E., “Decomposable branching processses with a fixed moment of extinction”, Proc. Steklov Inst.Mathem., 290 (2015) (to appear).10.1134/S0081543815060103Search in Google Scholar

[5] Foster J., Ney P., “Decomposable critical multi-type branching processes”, Sanhya: the Indian J. Statist., Ser. A, 38 (1976), 28–37.Search in Google Scholar

[6] Foster J., Ney P., “Limit laws for decomposable critical branching processes”, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 46 (1978), 13–43.10.1007/BF00535685Search in Google Scholar

[7] Markushevich A.I., The theory of analytic functions, Translated from the second Russian edition, Delhi: Hindustan Publishing Corp., 1963.Search in Google Scholar

[8] Ogura Y., “Asymptotic behavior of multitype Galton-Watson processes”, J. Math. Kyoto Univ., 15 (1975), 251–302.10.1215/kjm/1250523066Search in Google Scholar

[9] Polin A. K., “Limit theorems for decomposable critical branching processes”, Mathematics of the USSR-Sbornik, 29:3 (1976), 377–392.10.1070/SM1976v029n03ABEH003674Search in Google Scholar

[10] Polin A. K., “Limit theorems for decomposable branching processes with final types”, Mathematics of the USSR-Sbornik, 33:1 (1977), 136–146.10.1070/SM1977v033n01ABEH002417Search in Google Scholar

[11] Savin A. A., Chistyakov V. P., “Some theorems for branching processes with several types of particles”, Theory Probab. Appl., 7:1 (1962), 93–100.10.1137/1107008Search in Google Scholar

[12] Vatutin V. A., “The structure of decomposable reduced branching processes. I. Finite-dimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662.10.1137/S0040585X97T987326Search in Google Scholar

[13] Vatutin V. A., Sagitov S. M., “A decomposable critical branching process with two types of particles”, Proc. Steklov Inst. Math., 177 (1988), 1–19.Search in Google Scholar

[14] Vatutin V.A., Dyakonova E.E., Jagers P., Sagitov S.M., “A decomposable branching process in a markovian environment”, Article ID 694285, Int. J. Stoch. Analysis, 2012 (2012), 24 pp.10.1155/2012/694285Search in Google Scholar

[15] Zubkov A. M., “The limit behaviour of decomposable critical branching processes with two types of particles”, Theory Probab. Appl., 27:2 (1983), 235–237.10.1137/1127026Search in Google Scholar

Received: 2015-6-18
Published Online: 2016-7-15
Published in Print: 2016-7-1

© 2016 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/dma-2016-0016
Keywords for this article
decomposable branching processes; criticality; conditional limit theorems

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  1. Research Article
  2. Finite automata and numbers
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