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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(γz t), 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

Information

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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References

[1] Athreya, K. B. (1969) On the supercritical age-dependent branching process. Ann Math. Statist. 40, 743–763.Google Scholar
[2] Athreya, K. B. and Kaplan, N. (1976) Convergence of the age distribution in the one-dimensional supercritical age-dependent branching process. Ann. Prob. 4, 38–50.Google Scholar
[3] Athreya, K. B. and Kaplan, N. (1978) Additive property and its applications in branching processes. In Branching Processes ed. Joffe, A. and Ney, P. E., 27–60.Google Scholar
[4] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.Google Scholar
[5] Biggins, J. D. and Grey, D. R. (1979) Continuity of limit random variables in the branching random walk. J. Appl. Prob. 16, 740–749.Google Scholar
[6] Breiman, L. (1968) Probability. Addision-Wesley, Reading, Ma.Google Scholar
[7] Cohn, H. (1982) Norming constants for the finite mean supercritical Bellman-Harris process. Z. Wahrscheinlichkeitsth. CrossRefGoogle Scholar
[8] Cohn, H. and Schuh, H.-J. (1980) On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean. J. Appl. Prob. 17, 696–703.Google Scholar
[9] Feller, W. (1971) An Introduction to Probability and its Applications. Vol. II, 2nd edn. Wiley, New York.Google Scholar
[10] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[11] Heyde, C. C. (1970) Extension of a result by Seneta for the supercritical branching process. Ann. Math. Statist. 41, 739–742.CrossRefGoogle Scholar
[12] Hoppe, F. M. (1976) Supercritical multitype branching processes. Ann. Prob. 4, 393–401.Google Scholar
[13] Jagers, P. (1969) Renewal theory and the a.s. convergence of branching processes. Ark. Mat. 7, 495–504.CrossRefGoogle Scholar
[14] Kuczek, T. (1982) On the convergence of the empiric age distribution for one dimensional supercritical age-dependent branching processes. Ann. Prob. Google Scholar
[15] Kuczek, T. (1980) On Some Results in Branching Processes and GA/G/8 Queues with Applications to Biology. Ph.D. Thesis, Purdue University.Google Scholar
[16] Nerman, O. (1979) On the Convergence of Supercritical General Branching Process. Ph.D. Thesis, University of Goteborg.Google Scholar
[17] Schuh, H.-J. and Barbour, A. D. (1977) On the asymptotic behaviour of branching processses with infinite mean. Adv. Appl. Prob. 9, 681–723.Google Scholar
[18] Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39, 2098–2102.Google Scholar
[19] Seneta, E. (1975) Characterization by functional equations of branching process limit laws. In Statistical Distributions in Scientific Work, Vol. 3, ed. Patil, G. P. et al, Reidel, Dordrecht, 249–254.Google Scholar