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Seneta constants for the supercritical Bellman–Harris process

Published online by Cambridge University Press:  01 July 2016

H.-J. Schuh
H.-J. Schuh*
Affiliation:
University of Melbourne
*
∗Present address: Johannes Gutenberg-Universität in Mainz, Fachbereich 17, Mathematik, Saarstr. 21, Postfach 3980, D-6500 Mainz, W. Germany.
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Abstract

Let be a supercritical Bellman-Harris process with finite offspring mean. Cohn [17] has shown that there always exist constants Ct such that limt→∞Zt/Ct = W almost surely for some non-degenerate random variable W. In this paper we give an alternative proof, based on the study of (Zt) as a point process. Our methods are to some extent analytical and parallel Seneta's [18] and Heyde's [11] approaches in the case of the Galton–Watson process. We further identify Ct as 1/(–log Ft(–1)(γ)), where Ft(γ) = E(γzt), i.e. the norming constants found by Seneta [18] for the Galton–Watson process, apply also to the Bellman-Harris process. Finally we derive a weak law of large numbers for W, prove that W is continuous on (0,∞) and show that W has [0,∞) as its support.

Keywords

AGE-DEPENDENT BRANCHING PROCESSNORMING CONSTANTSMULTITYPE BRANCHING PROCESSPOINT PROCESSGENERAL BRANCHING PROCESSAGE DISTRIBUTIONMALTHUSIAN PARAMETERSEQUENCE OF BACKWARD ITERATESHEYDE-TYPE MARTINGALEASSOCIATED GENERATION PROCESSWEAK LAW OF LARGE NUMBERSCONTINUITY AND POSITIVITY OF THE DISTRIBUTION OF W
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 4 , December 1982 , pp. 732 - 751
Copyright
Copyright © Applied Probability Trust 1982 

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