## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 35, Number 2 (1964), 557-565.

### Limit Distributions of a Branching Stochastic Process

Howard H. Stratton, Jr. and Howard G. Tucker

#### Abstract

A population of particles is considered whose size $X_N(t)$ changes according to a branching stochastic process. The purpose of this paper is to find an approximate distribution of $X_N(t)$ when $t$ is fixed (not necessarily large) but the initial size of the population, $N$, is large. If $N$ is allowed to tend to infinity, and if the parameters of the process are made to change in a way analogous to the Poisson approximation of a binomial distribution, then it is shown that a limiting distribution of the process $X_N(t) - N$ exists as $N \rightarrow \infty$, and this limiting distribution is the distribution of a continuous process with independent increments. The relation between the parameters of the infinitely divisible distribution of the limiting process and the sequence of branching processes is exhibited.

#### Article information

**Source**

Ann. Math. Statist., Volume 35, Number 2 (1964), 557-565.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177703555

**Digital Object Identifier**

doi:10.1214/aoms/1177703555

**Mathematical Reviews number (MathSciNet)**

MR160272

**Zentralblatt MATH identifier**

0245.60064

**JSTOR**

links.jstor.org

#### Citation

Stratton, Howard H.; Tucker, Howard G. Limit Distributions of a Branching Stochastic Process. Ann. Math. Statist. 35 (1964), no. 2, 557--565. doi:10.1214/aoms/1177703555. https://projecteuclid.org/euclid.aoms/1177703555