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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2, Pages 274–300 (Mi tvp285)  

This article is cited in 22 scientific papers (total in 22 papers)

Galton–Watson branching processes in a random environment. I: limit theorems

V. A. Vatutin, E. E. D'yakonova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $Z_n$ be the number of individuals at time $n$ in a branching process in a random environment generated by independent identically distributed random probability generating functions $f_0(s),f_1(s),…,f_n(s),…$ . Let
$$ X_i=\log f_{i-1}'(1),\qquad i=0,1,2,…; \qquad S_0=0,\quad S_n=X_1+…+X_n,\qquad n\ge 1. $$
It is shown that if $Z_n$ is, in a sense, “critical,” then there exists a limit in distribution
$$ \lim_{n\to\infty}\exp\{-\min_{0\le j\le n}S_j\} \mathbf{P}Ż_n>0\mid f_0,…,f_{n-1}\}=\zeta, $$
where $\zeta$ is a proper random variable positive with probability 1. In addition, it is shown that for a “typical” realization of the environment the number of individuals $Z_n$ given $Ż_n>0\}$ grows as $\exp\{S_n-\min_{0\le j\le n}S_j\}$ (up to a positive finite random multiplier).

Keywords: branching processes in random environment, survival probability, critical branching process, random walks, stable distributions, harmonic functions.

DOI: https://doi.org/10.4213/tvp285

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English version:
Theory of Probability and its Applications, 2004, 48:2, 314–336

Bibliographic databases:

Received: 30.10.2002

Citation: V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. I: limit theorems”, Teor. Veroyatnost. i Primenen., 48:2 (2003), 274–300; Theory Probab. Appl., 48:2 (2004), 314–336

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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
    2. V. A. Vatutin, E. E. D'yakonova, “Galton–Watson branching processes in a random environment. II: Finite-dimensional distributions”, Theory Probab. Appl., 49:2 (2005), 275–309  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Afanasyev V.I., Geiger J., Kersting G., Vatutin V.A., “Criticality for branching processes in random environment”, Ann. Probab., 33:2 (2005), 645–673  crossref  mathscinet  zmath  isi  elib  scopus
    4. V. A. Vatutin, E. E. D'yakonova, “Branching processes in random environment and “bottlenecks” in evolution of populations”, Theory Probab. Appl., 51:1 (2007), 189–210  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. E. E. D'yakonova, “Critical multitype branching processes in a random environment”, Discrete Math. Appl., 17:6 (2007), 587–606  mathnet  crossref  crossref  mathscinet  zmath  elib
    6. 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
    7. 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
    8. V. A. Vatutin, E. E. Dyakonova, “Asymptotic properties of multitype critical branching processes evolving in a random environment”, Discrete Math. Appl., 20:2 (2010), 157–177  mathnet  crossref  crossref  mathscinet  elib
    9. V. A. Vatutin, “Polling systems and multitype branching processes in a random environment with final product”, Theory Probab. Appl., 55:4 (2011), 631–660  mathnet  crossref  crossref  mathscinet  isi
    10. V. A. Vatutin, “Multitype branching processes with immigration in random environment, and polling systems”, Siberian Adv. Math., 21:1 (2011), 42–72  mathnet  crossref  mathscinet  elib  elib
    11. E. E. D'yakonova, “Multitype Galton–Watson branching processes in Markovian random environment”, Theory Probab. Appl., 56:3 (2011), 508–517  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. V. A. Vatutin, Q. Liu, “Critical branching process with two types of particles evolving in asynchronous random environments”, Theory Probab. Appl., 57:2 (2013), 279–305  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. E. E. D'yakonova, “Multitype branching processes evolving in a Markovian environment”, Discrete Math. Appl., 22:5-6 (2012), 639–664  mathnet  crossref  crossref  mathscinet  elib
    14. V. I. Afanasyev, “About time of reaching a high level by a random walk in a random environment”, Theory Probab. Appl., 57:4 (2013), 547–567  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    15. 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
    16. E. E. Dyakonova, “Branching processes in a Markov random environment”, Discrete Math. Appl., 24:6 (2014), 327–343  mathnet  crossref  crossref  mathscinet  elib  elib
    17. Boeinghoff Ch., “Limit Theorems For Strongly and Intermediately Supercritical Branching Processes in Random Environment With Linear Fractional Offspring Distributions”, Stoch. Process. Their Appl., 124:11 (2014), 3553–3577  crossref  mathscinet  zmath  isi  scopus
    18. Afanasyev V.I. Boeinghoff Ch. Kersting G. Vatutin V.A., “Conditional Limit Theorems For Intermediately Subcritical Branching Processes in Random Environment”, Ann. Inst. Henri Poincare-Probab. Stat., 50:2 (2014), 602–627  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    19. E. E. D'yakonova, “Limit theorem for multitype critical branching process evolving in random environment”, Discrete Math. Appl., 25:3 (2015), 137–147  mathnet  crossref  crossref  mathscinet  isi  elib
    20. 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
    21. 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
    22. 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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