Critical Multitype Branching Processes

Abstract

A general multitype branching process in which individuals are counted according to some possibly type-dependent characteristic may be defined along the lines laid out by Jagers (1969, 1974) for the single type process. In the critical case, the probability of nonextinction at time $t$ is shown to be $O(t^{-1})$, and, conditioned on nonextinction at time $t$, the totals of the characteristic counts, normalized by $t$, are shown to satisfy an exponential limit law, under weak (essentially, second moment) hypotheses.