Branching Processes

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 pᵣ, r = 0, 1, 2,. 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 zₙ be the number of individuals in the nth generation. We study the probability distribution of zₙ. Some previous results are given in section 2; these include procedures for computing moments of zₙ, 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 zₙ/Ezₙ converge in probability to a random variable w with cumulative distribution G (u). It is shown that G (u) is absolutely continuous for u 0. Results of a Tauberian character are given for the behavior of G (u) as u 0 and u. In section 5 some examples are given where G (u) can be found explicitly; G (u) is computed numerically for the case p₁ = 0. 4, p₂ = 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 pᵣ and for the expectation Ez₁; 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.