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December, 1973 The Growth of Supercritical Branching Processes with Random Environments
Niels Keiding, John E. Nielsen
Ann. Probab. 1(6): 1065-1067 (December, 1973). DOI: 10.1214/aop/1176996814


For the supercritical branching process with random environments, the rate of growth of the generation size $Z_n$ is studied in the marginal distribution. It is shown that unless the environmental process yields a constant conditional expectation $E(Z_1 \mid \zeta)$, the asymptotic distribution of $$(Z_n \exp(-nE_\zeta(\log E(Z_1 \mid \zeta))))^{n^{-\frac{1}{2}}}$$ is that of $Ue^V$ where $U$ and $V$ are independent, $P(U = 0) = 1 - P(U = 1) = P(Z_n \rightarrow 0)$ and $V$ is normal $(0, V_\zeta(\log E(Z_1\mid \zeta))$.


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Niels Keiding. John E. Nielsen. "The Growth of Supercritical Branching Processes with Random Environments." Ann. Probab. 1 (6) 1065 - 1067, December, 1973.


Published: December, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0272.60057
MathSciNet: MR359045
Digital Object Identifier: 10.1214/aop/1176996814

Primary: 60J80

Keywords: Branching process with random environments

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 6 • December, 1973
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