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A note on simple branching processes with infinite mean

Published online by Cambridge University Press:  14 July 2016

Irene L. Hudson  and
E. Seneta
Irene L. Hudson
Affiliation:
Cambridge University
E. Seneta*
Affiliation:
Virginia Polytechnic Institute and State University
*
*Permanent address: The Australian National University, Canberra.
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Abstract

We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn} say, conditions are given ensuring that there exists a sequence of positive constants, {ρn}, such that {ρnU(Zn + 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.

Keywords

BRANCHING PROCESSIMMIGRATIONINFINITE MEANSLOWLY VARYING FUNCTION
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 836 - 842
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Cohn, H. (1977) Almost sure convergence of branching processes. Z. Wahrscheinlichkeitsth. 38, 73–81.Google Scholar
[2] Kuczma, M. (1967) Un théorème d'unicité pour l'équation fonctionelle de Böttcher. Mathematica (Cluj) 9, 285–293.Google Scholar
[3] Pakes, A. G. (1976) Some limit theorems for a supercritical branching process with immigration. J. Appl. Prob. 13, 17–26.Google Scholar
[4] Reuter, G. E. H. (1975) Private communication, 18 November 1975.Google Scholar
[5] Seneta, E. (1970) On the supercritical Galton-Watson process with immigration. Math. Biosci. 7, 9–14.Google Scholar
[6] Seneta, E. (1973) The simple branching process with infinite mean. I. J. Appl. Prob. 10, 206–212.Google Scholar
[7] Seneta, E. (1974) Regularly varying functions in the theory of simple branching processes. Adv. Appl. Prob. 6, 408–420.Google Scholar
[8] Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508, Springer, Berlin.Google Scholar
[9] Smith, W. L. (1967) Remarks on renewal theory when the quality of renewals varies. Inst. Statist. Mimeo Ser. 548. Univerisity of North Carolina.Google Scholar