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Diskr. Mat., 1999, Volume 11, Issue 2, Pages 86–102 (Mi dm369)  

This article is cited in 14 scientific papers (total in 14 papers)

On the maximum of 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. We demonstrate that, under some conditions, as $x\to\infty$,
$$ \mathsf P(\sup_n\xi_n>x)\sim \frac{c_0}{\ln x},\qquad \mathsf P(\sum_{n=0}^\infty\xi_n>x)\sim \frac{c_0}{\ln x}, $$
where $c_0$ is a positive constant.

DOI: https://doi.org/10.4213/dm369

Full text: PDF file (1027 kB)

English version:
Discrete Mathematics and Applications, 1999, 9:3, 267–284

Bibliographic databases:

UDC: 519.2
Received: 03.07.1998

Citation: V. I. Afanasyev, “On the maximum of a critical branching process in a random environment”, Diskr. Mat., 11:2 (1999), 86–102; Discrete Math. Appl., 9:3 (1999), 267–284

Citation in format AMSBIB
\Bibitem{Afa99}
\by V.~I.~Afanasyev
\paper On the maximum of a critical branching process in a random environment
\jour Diskr. Mat.
\yr 1999
\vol 11
\issue 2
\pages 86--102
\mathnet{http://mi.mathnet.ru/dm369}
\crossref{https://doi.org/10.4213/dm369}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1712160}
\zmath{https://zbmath.org/?q=an:0977.60089}
\transl
\jour Discrete Math. Appl.
\yr 1999
\vol 9
\issue 3
\pages 267--284


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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. Afanasyev V.I., “On the maximum of a subcritical branching process in a random environment”, Stochastic Processes and Their Applications, 93:1 (2001), 87–107  crossref  mathscinet  zmath  isi  scopus
    3. 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
    4. 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
    5. V. A. Vatutin, “Total Population Size in Critical Branching Processes in a Random Environment”, Math. Notes, 91:1 (2012), 12–21  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    6. 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
    7. 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
    8. E. E. Dyakonova, “Branching processes in a Markov random environment”, Discrete Math. Appl., 24:6 (2014), 327–343  mathnet  crossref  crossref  mathscinet  elib  elib
    9. Vatutin V., Liu Q., “Limit Theorems For Decomposable Branching Processes in a Random Environment”, J. Appl. Probab., 52:3 (2015), 877–893  crossref  mathscinet  zmath  isi  elib
    10. 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
    11. 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
    12. Aurzada F., Devulder A., Guillotin-Plantard N., Pene F., “Random Walks and Branching Processes in Correlated Gaussian Environment”, J. Stat. Phys., 166:1 (2017), 1–23  crossref  mathscinet  zmath  isi  scopus
    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
    14. Chen X., Guillotin-Plantard N., “Branching Processes in Correlated Random Environment”, Electron. Commun. Probab., 24 (2019), 71  crossref  isi
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