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Upper large deviations of branching processes in a random environment--Offspring distributions with geometrically bounded tails

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  • Böinghoff, Christian
  • Kersting, Götz

Abstract

We generalize a result by Kozlov on large deviations of branching processes (Zn) in an i.i.d. random environment. Under the assumption that the offspring distributions have geometrically bounded tails and mild regularity of the associated random walk S, the asymptotics of is (on logarithmic scale) completely determined by a convex function [Gamma] depending on properties of S. In many cases [Gamma] is identical with the rate function of (Sn). However, if the branching process is strongly subcritical, there is a phase transition and the asymptotics of and differ for small [theta].

Suggested Citation

  • Böinghoff, Christian & Kersting, Götz, 2010. "Upper large deviations of branching processes in a random environment--Offspring distributions with geometrically bounded tails," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2064-2077, September.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:10:p:2064-2077
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    References listed on IDEAS

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    1. Afanasyev, V.I. & Geiger, J. & Kersting, G. & Vatutin, V.A., 2005. "Functional limit theorems for strongly subcritical branching processes in random environment," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1658-1676, October.
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    Cited by:

    1. Nakashima, Makoto, 2013. "Lower deviations of branching processes in random environment with geometrical offspring distributions," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3560-3587.
    2. Grama, Ion & Liu, Quansheng & Miqueu, Eric, 2017. "Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1255-1281.
    3. Huang, Chunmao & Liu, Quansheng, 2012. "Moments, moderate and large deviations for a branching process in a random environment," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 522-545.

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