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 Diskr. Mat., 2000, Volume 12, Issue 2, Pages 31–50 (Mi dm326)

On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk

V. I. Afanasyev

Abstract: Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions, $T$ be the time of extinction of $\{\xi_n\}$, $T_M$ be the first maximum passage time of $\{\xi_n\}$. We study the asymptotic behaviour of $\mathsf P(T_M>n)$ and prove limit theorems for the random variables $\{T_M/n\mid T>n\}$ and $\{T_M/T\mid T>n\}$ as $n\to\infty$. Similar results are established for the stopped random walk with zero drift.

DOI: https://doi.org/10.4213/dm326

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English version:
Discrete Mathematics and Applications, 2000, 10:3, 243–264

Bibliographic databases:

UDC: 519.2

Citation: 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”, Diskr. Mat., 12:2 (2000), 31–50; Discrete Math. Appl., 10:3 (2000), 243–264

Citation in format AMSBIB
\Bibitem{Afa00} \by V.~I.~Afanasyev \paper On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk \jour Diskr. Mat. \yr 2000 \vol 12 \issue 2 \pages 31--50 \mathnet{http://mi.mathnet.ru/dm326} \crossref{https://doi.org/10.4213/dm326} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1783073} \zmath{https://zbmath.org/?q=an:0969.60087} \transl \jour Discrete Math. Appl. \yr 2000 \vol 10 \issue 3 \pages 243--264 

• http://mi.mathnet.ru/eng/dm326
• https://doi.org/10.4213/dm326
• http://mi.mathnet.ru/eng/dm/v12/i2/p31

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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. I. Afanasyev, “Functional limit theorem for a stopped random walk attaining a high level”, Discrete Math. Appl., 27:5 (2017), 269–276
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