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\(\mathop {\lim }\limits_{t \to \infty } {{Z(t)} \mathord{\left/ {\vphantom {{Z(t)} {C(t)}}} \right. \kern-\nulldelimiterspace} {C(t)}} = W\) a.s. for some nondegenerate random variableW. C(t) is theμ-quantile of the distribution function ofZ(t), whereq<μ<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)}.
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Cohn, H. Norming constants for the finite mean supercritical Bellman-Harris process. Z. Wahrscheinlichkeitstheorie verw Gebiete 61, 189–205 (1982). https://doi.org/10.1007/BF01844631
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DOI: https://doi.org/10.1007/BF01844631