## The Annals of Probability

### The Growth of Supercritical Branching Processes with Random Environments

#### 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 $(0, V_\zeta(\log E(Z_1\mid \zeta))$.

#### Article information

Source
Ann. Probab., Volume 1, Number 6 (1973), 1065-1067.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996814

Digital Object Identifier
doi:10.1214/aop/1176996814

Mathematical Reviews number (MathSciNet)
MR359045

Zentralblatt MATH identifier
0272.60057

JSTOR