Moments, moderate and large deviations for a branching process in a random environment
Let (Zn) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Zn/E[Zn|ξ]. We show large and moderate deviation principles for the sequence logZn (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Zn. Central limit theorems on W−Wn and logZn are also established.
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Volume (Year): 122 (2012)
Issue (Month): 2 ()
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References listed on IDEAS
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- Liu, Quansheng, 1999. "Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 61-87, July.
- Liu, Quansheng, 2001. "Asymptotic properties and absolute continuity of laws stable by random weighted mean," Stochastic Processes and their Applications, Elsevier, vol. 95(1), pages 83-107, September.
- 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.
- Wang, Hesong & Gao, Zhiqiang & Liu, Quansheng, 2011. "Central limit theorems for a supercritical branching process in a random environment," Statistics & Probability Letters, Elsevier, vol. 81(5), pages 539-547, May.
- 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.
- Tanny, David, 1988. "A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means," Stochastic Processes and their Applications, Elsevier, vol. 28(1), pages 123-139, April.
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