Limit Distributions of a Branching Stochastic Process

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.