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Diskr. Mat., 1999, Volume 11, Issue 4, Pages 33–47 (Mi dm390)  

This article is cited in 10 scientific papers (total in 10 papers)

On the time of reaching a fixed level by a critical branching process in a random environment

V. I. Afanasyev


Abstract: Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions; let $T$ be the extinction time of $\{\xi_n\}$, and $T_x$ be the time of first passage of the semiaxis $(x,\infty)$. We find the asymptotic distributions of the random variables $T_x/\ln^2 x$, $T_x/T$, $T/\ln^2x$ under the condition $\{T_x<\infty\}$ as $x\to \infty$.

DOI: https://doi.org/10.4213/dm390

Full text: PDF file (937 kB)

English version:
Discrete Mathematics and Applications, 1999, 9:6, 627–643

Bibliographic databases:

UDC: 519.2
Received: 03.07.1998

Citation: V. I. Afanasyev, “On the time of reaching a fixed level by a critical branching process in a random environment”, Diskr. Mat., 11:4 (1999), 33–47; Discrete Math. Appl., 9:6 (1999), 627–643

Citation in format AMSBIB
\Bibitem{Afa99}
\by V.~I.~Afanasyev
\paper On the time of reaching a fixed level by a critical branching process in a random environment
\jour Diskr. Mat.
\yr 1999
\vol 11
\issue 4
\pages 33--47
\mathnet{http://mi.mathnet.ru/dm390}
\crossref{https://doi.org/10.4213/dm390}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1761010}
\zmath{https://zbmath.org/?q=an:0971.60087}
\transl
\jour Discrete Math. Appl.
\yr 1999
\vol 9
\issue 6
\pages 627--643


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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, “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
    3. V. I. Afanasyev, “Arcsine law for branching processes in a random environment and Galton–Watson processes”, Theory Probab. Appl., 51:3 (2007), 401–414  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. V. I. Afanasyev, “Invariance Principle for the Critical Branching Process in a Random Environment Attaining a High Level”, Theory Probab. Appl., 54:1 (2010), 1–13  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. V. I. Afanasyev, “Brownian high jump”, Theory Probab. Appl., 55:2 (2011), 183–197  mathnet  crossref  crossref  mathscinet  isi
    6. Afanasyev V.I., “New Invariance Principles for Critical Branching Process in Random Environment”, Advances in Data Analysis - Theory and Applications to Reliability and Inference, Data Mining, Bioinformatics, Lifetime Data, and Neural Networks, Statistics for Industry and Technology, 2010, 105–115  mathscinet  isi
    7. V. I. Afanasyev, “High level subcritical branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 4–14  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. V. I. Afanasyev, “Conditional limit theorem for maximum of random walk in a random environment”, Theory Probab. Appl., 58:4 (2014), 525–545  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. V. I. Afanasyev, “Functional limit theorem for a stopped random walk attaining a high level”, Discrete Math. Appl., 27:5 (2017), 269–276  mathnet  crossref  crossref  mathscinet  isi  elib
    10. 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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