The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 35, Number 2 (1964), 557-565.
Limit Distributions of a Branching Stochastic Process
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.
Ann. Math. Statist., Volume 35, Number 2 (1964), 557-565.
First available in Project Euclid: 27 April 2007
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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