## The Annals of Probability

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

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

Niels Keiding and John E. Nielsen

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Branching process with random environments

#### Citation

Keiding, Niels; Nielsen, John E. The Growth of Supercritical Branching Processes with Random Environments. Ann. Probab. 1 (1973), no. 6, 1065--1067. doi:10.1214/aop/1176996814. https://projecteuclid.org/euclid.aop/1176996814