A limit theorem for a class of supercritical branching processes

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes { ξ n } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E { ξ 1 } . Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z ( t )/ E [ Z ( t )], as t → ∞, where Z ( t ) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z ( t )/ E { Z ( t )} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξ n / m n to have a non-degenerate limit. An integral equation is established for the generating function of Z ( t ), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N ( x ), x 0, ≧ the number of children born to an individual in the age interval [0, x ] .