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 p 0 , p 1 , p 2 , …. 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.