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 Diskr. Mat., 2001, Volume 13, Issue 4, Pages 73–91 (Mi dm300)

A functional limit theorem for a critical branching process in a random environment

V. I. Afanasyev

Abstract: Let $\{\xi_n\}$ be a critical branching process in a random environment, and let $m_n$ be the mathematical expectation of $\xi_n$ under the condition that the random environment is fixed. We prove a theorem on convergence of the sequence of branching processes $\{\xi_{[nt]}/m_{[nt]}, t\in(0,1] \mid \xi_n>0\}$ as $n\to\infty$ in distribution in the corresponding functional space. This theorem extends the earlier result of the author proved under the assumption that the generating function of the number of offspring is linear-fractional.

DOI: https://doi.org/10.4213/dm300

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English version:
Discrete Mathematics and Applications, 2001, 11:6, 587–606

Bibliographic databases:

UDC: 519.2

Citation: V. I. Afanasyev, “A functional limit theorem for a critical branching process in a random environment”, Diskr. Mat., 13:4 (2001), 73–91; Discrete Math. Appl., 11:6 (2001), 587–606

Citation in format AMSBIB
\Bibitem{Afa01} \by V.~I.~Afanasyev \paper A~functional limit theorem for a critical branching process in a random environment \jour Diskr. Mat. \yr 2001 \vol 13 \issue 4 \pages 73--91 \mathnet{http://mi.mathnet.ru/dm300} \crossref{https://doi.org/10.4213/dm300} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1901784} \zmath{https://zbmath.org/?q=an:1102.60306} \transl \jour Discrete Math. Appl. \yr 2001 \vol 11 \issue 6 \pages 587--606 

• http://mi.mathnet.ru/eng/dm300
• https://doi.org/10.4213/dm300
• http://mi.mathnet.ru/eng/dm/v13/i4/p73

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. I. Afanasyev, “On the ratio between the maximal and total numbers of individuals in a critical branching process in a random environment”, Theory Probab. Appl., 48:3 (2004), 384–399
2. V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. I: limit theorems”, Theory Probab. Appl., 48:2 (2004), 314–336
3. Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Criticality for branching processes in random environment”, Annals of Probability, 33:2 (2005), 645–673
4. V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242
5. Vatutin V. Dyakonova E., “Path to Survival For the Critical Branching Processes in a Random Environment”, J. Appl. Probab., 54:2 (2017), 588–602
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