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Lower deviations of branching processes in random environment with geometrical offspring distributions

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  • Nakashima, Makoto

Abstract

We consider the branching processes in random environment. In this paper, we deal with the case of environments which are chosen stationary and ergodic from the finite set of geometrical offspring distributions. We denote by Zn the population at the n-th generation. We show that the large deviation principle holds with a certain rate function for the total population when the environment satisfies some conditions. Also, we will show that the trajectory t→logZntn,t∈[0,1] converges to a deterministic function uniformly in probability conditioned on {0

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  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:9:p:3560-3587
    DOI: 10.1016/j.spa.2013.04.013
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Böinghoff, Christian, 2014. "Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3553-3577.
    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.

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