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

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

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 

• http://mi.mathnet.ru/eng/dm390
• https://doi.org/10.4213/dm390
• http://mi.mathnet.ru/eng/dm/v11/i4/p33

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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
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
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
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
5. V. I. Afanasyev, “Brownian high jump”, Theory Probab. Appl., 55:2 (2011), 183–197
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
7. V. I. Afanasyev, “High level subcritical branching processes in a random environment”, Proc. Steklov Inst. Math., 282 (2013), 4–14
8. V. I. Afanasyev, “Conditional limit theorem for maximum of random walk in a random environment”, Theory Probab. Appl., 58:4 (2014), 525–545
9. V. I. Afanasyev, “Functional limit theorem for a stopped random walk attaining a high level”, Discrete Math. Appl., 27:5 (2017), 269–276
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
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