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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

Information

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

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