## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 3 (1974), 509-514.

### A Note on the Supercritical Branching Processes with Random Environments

#### Abstract

Some further results in the theory of Galton Watson processes are extended to the more general set up of a branching process with random environments. The random distribution function of the limit random variable in the supercritical case (Athreya and Karlin, Ann. Math. Statist., 40 (1969) 743-763) is investigated, and a zero-one law is established. It is shown that this random distribution function is w.p. 1. either absolutely continuous on $(0, \infty)$ with only a jump at the origin or w.p. 1. it is singular. A set of conditions is given under which the former case holds.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 3 (1974), 509-514.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996668

**Digital Object Identifier**

doi:10.1214/aop/1176996668

**Mathematical Reviews number (MathSciNet)**

MR356265

**Zentralblatt MATH identifier**

0293.60079

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J85: Applications of branching processes [See also 92Dxx]

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

**Keywords**

Branching process branching process with random environment random environment stationary ergodic process supercritical branching process

#### Citation

Kaplan, Norman. A Note on the Supercritical Branching Processes with Random Environments. Ann. Probab. 2 (1974), no. 3, 509--514. doi:10.1214/aop/1176996668. https://projecteuclid.org/euclid.aop/1176996668