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Diskr. Mat., 1997, Volume 9, Issue 3, Pages 52–67 (Mi dm479)  

This article is cited in 13 scientific papers (total in 13 papers)

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

V. I. Afanasyev


Abstract: Let $\{\xi_n\}$ be a critical branching process in random environment with linear-fractional generating functions, $m_n$ be the conditional expectation of $\xi_n$ with respect to random environment. We prove a theorem on convergence of the sequence of random processes $\{\xi_{[nt]}/m_{[nt]}, t\in(0,1] \mid \xi_n>0\}$ as $n\to\infty$ in distribution in the corresponding functional space.

DOI: https://doi.org/10.4213/dm479

Full text: PDF file (1180 kB)

English version:
Discrete Mathematics and Applications, 1997, 7:5, 497–513

Bibliographic databases:

UDC: 519.2
Received: 09.02.1996
Revised: 21.05.1997

Citation: V. I. Afanasyev, “A new limit theorem for a critical branching process in a random environment”, Diskr. Mat., 9:3 (1997), 52–67; Discrete Math. Appl., 7:5 (1997), 497–513

Citation in format AMSBIB
\Bibitem{Afa97}
\by V.~I.~Afanasyev
\paper A new limit theorem for a critical branching process in a random environment
\jour Diskr. Mat.
\yr 1997
\vol 9
\issue 3
\pages 52--67
\mathnet{http://mi.mathnet.ru/dm479}
\crossref{https://doi.org/10.4213/dm479}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1485649}
\zmath{https://zbmath.org/?q=an:0969.60085}
\transl
\jour Discrete Math. Appl.
\yr 1997
\vol 7
\issue 5
\pages 497--513


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. I. Afanasyev, “On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk”, Discrete Math. Appl., 10:3 (2000), 243–264  mathnet  crossref  mathscinet  zmath
    2. V. I. Afanasyev, “Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment”, Discrete Math. Appl., 11:2 (2001), 105–131  mathnet  crossref  mathscinet  zmath
    3. V. I. Afanasyev, “A functional limit theorem for a critical branching process in a random environment”, Discrete Math. Appl., 11:6 (2001), 587–606  mathnet  crossref  mathscinet  zmath
    4. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. II: Finite-dimensional distributions”, Theory Probab. Appl., 49:2 (2005), 275–309  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. 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  crossref  mathscinet  zmath  isi  scopus
    8. V. I. Afanasyev, “About time of reaching a high level by a random walk in a random environment”, Theory Probab. Appl., 57:4 (2013), 547–567  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. V. A. Vatutin, E. E. Dyakonova, S. Sagitov, “Evolution of branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 220–242  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    10. E. E. Dyakonova, “Branching processes in a Markov random environment”, Discrete Math. Appl., 24:6 (2014), 327–343  mathnet  crossref  crossref  mathscinet  elib  elib
    11. V. I. Afanasyev, “On the time of attaining a high level by a transient random walk in a random environment”, Theory Probab. Appl., 61:2 (2017), 178–207  mathnet  crossref  crossref  mathscinet  isi  elib
    12. V. I. Afanasyev, “On the non-recurrent random walk in a random environment”, Discrete Math. Appl., 28:3 (2018), 139–156  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Vatutin V. Dyakonova E., “Path to Survival For the Critical Branching Processes in a Random Environment”, J. Appl. Probab., 54:2 (2017), 588–602  crossref  mathscinet  isi  scopus
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