The number of individuals in a cascade process

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.