Abstract
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 $(0, V_\zeta(\log E(Z_1\mid \zeta))$.
Citation
Niels Keiding. John E. Nielsen. "The Growth of Supercritical Branching Processes with Random Environments." Ann. Probab. 1 (6) 1065 - 1067, December, 1973. https://doi.org/10.1214/aop/1176996814
Information