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Diskr. Mat., 2006, Volume 18, Issue 2, Pages 29–47 (Mi dm44)  

This article is cited in 15 scientific papers (total in 15 papers)

On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants

M. V. Kozlov


Abstract: A branching process $Z_n$ with geometric distribution of descendants in a random environment represented by a sequence of independent identically distributed random variables (the Smith–Wilkinson model) is considered. The asymptotics of large deviation probabilities $\boldsymbol{\mathsf P}(\ln Z_n>\theta n)$, $\theta>0$, are found provided that the steps of the accompanying random walk $S_n$ satisfy the Cramér condition. In the cases of supercritical, critical, moderate, and intermediate subcritical processes the asymptotics follow that of the large deviations probabilities $\boldsymbol{\mathsf P}(S_n\le\theta n)$. In strongly subcritical case the same asymptotics hold for $\theta$ greater than some $\theta^*$ (for $\theta\le\theta^*$ the asymptotics of large deviation probabilities are different).
This research was supported by the Russian Foundation for Basic Research, grant 04–01–00700, and by DFG, project 436 RUS 113/722.

DOI: https://doi.org/10.4213/dm44

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English version:
Discrete Mathematics and Applications, 2006, 16:2, 155–174

Bibliographic databases:

UDC: 519.2
Received: 02.11.2004
Revised: 07.04.2006

Citation: M. V. Kozlov, “On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants”, Diskr. Mat., 18:2 (2006), 29–47; Discrete Math. Appl., 16:2 (2006), 155–174

Citation in format AMSBIB
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\paper On large deviations of branching processes in a random environment: a geometric distribution of the number of descendants
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\issue 2
\pages 29--47
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\zmath{https://zbmath.org/?q=an:1126.60089}
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\jour Discrete Math. Appl.
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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. M. V. Kozlov, “On large deviations of strictly critical branching processes in a random environment with geometric distribution of descendants”, Theory Probab. Appl., 54:3 (2010), 424–446  mathnet  crossref  crossref  mathscinet  isi
    2. Böinghoff Ch., Kersting G., “Upper large deviations of branching processes in a random environment-Offspring distributions with geometrically bounded tails”, Stochastic Process. Appl., 120:10 (2010), 2064–2077  crossref  mathscinet  zmath  isi  scopus
    3. Wang W.G., “Bounds of deviation for branching chains in random environments”, Acta Math. Sin. (Engl. Ser.), 27:5 (2011), 897–904  mathscinet  zmath  isi
    4. Bansaye V., Böinghoff Ch., “Upper large deviations for branching processes in random environment with heavy tails”, Electron. J. Probab., 16:69 (2011), 1900–1933  crossref  mathscinet  zmath  isi  elib  scopus
    5. Huang Ch., Liu Q., “Moments, moderate and large deviations for a branching process in a random environment”, Stochastic Process. Appl., 122:2 (2012), 522–545  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. V. Shklyaev, “On large deviations of branching processes in a random environment with arbitrary initial number of particles: critical and supercritical cases”, Discrete Math. Appl., 22:5-6 (2012), 619–638  mathnet  crossref  crossref  mathscinet  elib
    7. Proc. Steklov Inst. Math., 282 (2013), 15–34  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. Proc. Steklov Inst. Math., 282 (2013), 45–61  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. Nakashima M., “Lower Deviations of Branching Processes in Random Environment with Geometrical Offspring Distributions”, Stoch. Process. Their Appl., 123:9 (2013), 3560–3587  crossref  mathscinet  zmath  isi  scopus
    10. D. V. Dmitrushchenkov, “On large deviations of a branching process in random environments with immigration at moments of extinction”, Discrete Math. Appl., 25:6 (2015), 339–343  mathnet  crossref  crossref  mathscinet  isi  elib
    11. Huang Ch., Liu Q., “Convergence in l-P and Its Exponential Rate For a Branching Process in a Random Environment”, Electron. J. Probab., 19 (2014), 104  crossref  mathscinet  zmath  isi  scopus
    12. D. V. Dmitrushchenkov, A. V. Shklyaev, “Large deviations of branching processes with immigration in random environment”, Discrete Math. Appl., 27:6 (2017), 361–376  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Grama I., Liu Q., Miqueu E., “Berry–Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment”, Stoch. Process. Their Appl., 127:4 (2017), 1255–1281  crossref  mathscinet  zmath  isi  scopus
    14. Grama I., Liu Q., Miqueu E., “Harmonic Moments and Large Deviations For a Supercritical Branching Process in a Random Environment”, Electron. J. Probab., 22 (2017), 99  crossref  mathscinet  zmath  isi  scopus
    15. A. V. Shklyaev, “Bolshie ukloneniya vetvyaschegosya protsessa v sluchainoi srede. II”, Diskret. matem., 32:1 (2020), 135–156  mathnet  crossref
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