Norming constants for the finite mean supercritical Bellman-Harris process

Summary Let {Z(t)} be a supercritical Bellman-Harris process with offspring distribution {p_k} and lifetime distributionG. It is shown that the finiteness of the offspring mean guarantees the existence of norming constants {C(t)} such that a.s. for some nondegenerate random variableW. C(t) is the-quantile of the distribution function ofZ(t), whereq<<1,q being the extinction probability of the process. As a byproduct of the proof, {Z(t)/C(t)} is shown to be asymptotic Markov. The theory of weakly stable sums of i.i.d. is used to get characterizations ofW and {C(t)}.