Branching Processes

Abstract

This paper is concerned with a simple mathematical model for a branching stochastic process. Using the language of family trees we may illustrate the process as follows. The probability that a man has exactly $r$ sons is $p_r, r = 0, 1, 2, \cdots$. Each of his sons (who together make up the first generation) has the same probabilities of having a given number of sons of his own; the second generation have again the same probabilities, and so on. Let $z_n$ be the number of individuals in the $n$th generation. We study the probability distribution of $z_n$. Some previous results are given in section 2; these include procedures for computing moments of $z_n$, and a criterion for when the family has probability 1 of dying out. In sections 3 and 4 the case is considered where the family has a non-zero chance of surviving indefinitely. In this case the random variables $z_n/Ez_n$ converge in probability to a random variable $w$ with cumulative distribution $G(u)$. It is shown that $G(u)$ is absolutely continuous for $u \neq 0$. Results of a Tauberian character are given for the behavior of $G(u)$ as $u \rightarrow 0$ and $u \rightarrow \infty$. In section 5 some examples are given where $G(u)$ can be found explicitly; $G(u)$ is computed numerically for the case $p_1 = 0.4, p_2 = 0.6$. In section 6 families with probability 1 of extinction are considered. A method is given for obtaining in certain cases an expansion for the moment-generating function of the number of generations before extinction occurs. In section 7 maximum likelihood estimates are obtained for the $p_r$ and for the expectation $Ez_1$; consistency in a certain sense is proved. In section 8 a brief discussion is given of the relation between two types of mathematical models for branching processes.