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 Diskr. Mat., 2001, Volume 13, Issue 1, Pages 132–157 (Mi dm270)

Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment

V. I. Afanasyev

Abstract: Let $\{\xi_n\}$ be an intermediately subcritical branching process in a random environment with linear-fractional generating functions, and let $m_n^+$ be the conditional mathematical expectation of $\xi_n$ under the condition that the random environment is fixed and $\xi_n>0$. We establish the convergence of the sequence of processes $\{\xi_{[nt]}/m^+_{[nt]}, t\in(0,1)\mid \xi_n>\nobreak0\}$ as $n\to\infty$ in the sense of finite-dimensional distributions. As a corollary, we establish the convergence of the sequence of processes $\{\ln\xi_{[nt]}/ \sqrt n, t\in[0,1]\mid \xi_n>0\}$ in the sense of finite-dimensional distributions to a process expressed in terms of the Brownian meander.
For a strongly subcritical branching process in a random environment $\{\xi_n\}$ with linear-fractional generating functions, we establish the convergence of the sequence $\{\xi_{[nt]}, t\in(0,1)\mid \xi_n>0\}$ in the sense of finite-dimensional distributions to a process whose all cross-sections are independent and identically distributed.
This research was supported by the Russian Foundation for Basic Research, grant 98–01–00524, and INTAS, grant 99–01317.

DOI: https://doi.org/10.4213/dm270

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English version:
Discrete Mathematics and Applications, 2001, 11:2, 105–131

Bibliographic databases:

UDC: 519.2

Citation: V. I. Afanasyev, “Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment”, Diskr. Mat., 13:1 (2001), 132–157; Discrete Math. Appl., 11:2 (2001), 105–131

Citation in format AMSBIB
\Bibitem{Afa01} \by V.~I.~Afanasyev \paper Limit theorems for an intermediately subcritical and a strongly subcritical branching process in a random environment \jour Diskr. Mat. \yr 2001 \vol 13 \issue 1 \pages 132--157 \mathnet{http://mi.mathnet.ru/dm270} \crossref{https://doi.org/10.4213/dm270} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1846044} \zmath{https://zbmath.org/?q=an:1045.60087} \transl \jour Discrete Math. Appl. \yr 2001 \vol 11 \issue 2 \pages 105--131 

• http://mi.mathnet.ru/eng/dm270
• https://doi.org/10.4213/dm270
• http://mi.mathnet.ru/eng/dm/v13/i1/p132

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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, “A functional limit theorem for a critical branching process in a random environment”, Discrete Math. Appl., 11:6 (2001), 587–606
2. V. A. Vatutin, “Limit theorem for an intermediate subcritical branching process in a random environment”, Theory Probab. Appl., 48:3 (2004), 481–492
3. Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Functional limit theorems for strongly subcritical branching processes in random environment”, Stochastic Processes and Their Applications, 115:10 (2005), 1658–1676
4. V. I. Afanasev, “Sluchainye bluzhdaniya i vetvyaschiesya protsessy”, Lekts. kursy NOTs, 6, MIAN, M., 2007, 3–187
5. Vatutin V. Zheng X., “Subcritical Branching Processes in a Random Environment Without the Cramer Condition”, Stoch. Process. Their Appl., 122:7 (2012), 2594–2609
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
7. Proc. Steklov Inst. Math., 282 (2013), 45–61
8. Vatutin V., “Subcritical Branching Processes in Random Environment”, Branching Processes and Their Applications, Lecture Notes in Statistics, 219, eds. DelPuerto I., Gonzalez M., Gutierrez C., Martinez R., Minuesa C., Molina M., Mota M., Ramos A., Springer, 2016, 97–115
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