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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 20.01.2000

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


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  • http://mi.mathnet.ru/eng/dm/v13/i1/p132

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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, “A functional limit theorem for a critical branching process in a random environment”, Discrete Math. Appl., 11:6 (2001), 587–606  mathnet  crossref  mathscinet  zmath
    2. V. A. Vatutin, “Limit theorem for an intermediate subcritical branching process in a random environment”, Theory Probab. Appl., 48:3 (2004), 481–492  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  scopus
    4. V. I. Afanasev, “Sluchainye bluzhdaniya i vetvyaschiesya protsessy”, Lekts. kursy NOTs, 6, MIAN, M., 2007, 3–187  mathnet  crossref  zmath  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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. Proc. Steklov Inst. Math., 282 (2013), 45–61  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  scopus
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