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Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 1, Pages 21–38 (Mi tvp2959)  

This article is cited in 10 scientific papers (total in 10 papers)

Reduced branching processes in random environment: the critical case

V. A. Vatutin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $Z_n$ be the number of particles at time $n=0,1,2,…$ in a branching process in random environment, $Z_0=1$, and let $Z_{m,n}$ be the number of such particles in the process at time $m\in[0,n]$, each of which has a nonempty offspring at time $n$. It is shown that if the offspring generating functions $f_k(s)$ of the particles of the $k$th generation are independent and identically distributed for all $k=0,1,2,…$ with $E\log f'_k(1)=0$ and $\sigma^2=E(\log f'_k(1))^2\in(0,\infty)$, then, under certain additional restrictions, the sequence of conditional processes
$$ \{\frac1{\sigma\sqrt{n}} \log Z_{[nt],n}, t\in[0,1]|Z_n>0\} $$
converges, as $n\to\infty$, in distribution in Skorokhod topology to the process $\{\inf_{t\le u\le 1}W^+(u), t\in[0,1]\}$, where $\{W_+(t), t\in [0,1]\}$ is the Brownian meander.

Keywords: critical branching process in random environment, reduced process, functional limit theorem, random walk.

DOI: https://doi.org/10.4213/tvp2959

Full text: PDF file (1421 kB)

English version:
Theory of Probability and its Applications, 2003, 47:1, 99–113

Bibliographic databases:

Received: 27.08.2001

Citation: V. A. Vatutin, “Reduced branching processes in random environment: the critical case”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 21–38; Theory Probab. Appl., 47:1 (2003), 99–113

Citation in format AMSBIB
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\jour Theory Probab. Appl.
\yr 2003
\vol 47
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\pages 99--113
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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. Vladimir V., Elena D., “Reduced branching processes in random environment”, Mathematics and Computer Science II - Algorithms, Trees, Combinatorics and Probabilities, Trends in Mathematics, 2002, 455–467  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. I: limit theorems”, Theory Probab. Appl., 48:2 (2004), 314–336  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. V. A. Vatutin, E. E. D'yakonova, “Limit theorems for reduced branching processes in a random environment”, Theory Probab. Appl., 52:2 (2008), 277–302  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. A. Vatutin, E. E. D'yakonova, “Waves in Reduced Branching Processes in a Random Environment”, Theory Probab. Appl., 53:4 (2009), 679–695  mathnet  crossref  crossref  mathscinet  zmath  isi
    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. A. Vatutin, “The structure of decomposable reduced branching processes. I. Finitedimensional distributions”, Theory Probab. Appl., 59:4 (2015), 641–662  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22  mathnet  crossref  crossref  mathscinet  isi  elib
    9. V. A. Vatutin, E. E. D'yakonova, “How many families survive for a long time?”, Theory Probab. Appl., 61:4 (2017), 692–711  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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