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On the continuity and the positivity of the finite part of the limit distribution of an irregular branching process with infinite mean

Published online by Cambridge University Press:  14 July 2016

H. Cohn  and
H.-J. Schuh
H. Cohn*
Affiliation:
University of Melbourne
H.-J. Schuh*
Affiliation:
University of Melbourne
*
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, Victoria 3052, Australia.
∗Postal address: Department of Statistics, Richard Berry Building, University of Melbourne, Parkville, Victoria 3052, Australia.
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Abstract

It is shown that the limiting random variable W(si) of an irregular branching process with infinite mean, defined in [5], has a continuous and positive distribution on {0 < W(si) < ∞}. This implies that for all branching processes (Zn) with infinite mean there exists a function U such that the distribution of V = limnU(Zn)e–n a.s. is continuous, positive and finite on the set of non-extinction. A kind of law of large numbers for sequences of independent copies of W(si) is derived.

Keywords

REGULAR AND IRREGULAR BRANCHING PROCESSES WITH INFINITE MEANREGULAR AND IRREGULAR POINTSCONTINUITY AND POSITIVITY OF A DISTRIBUTIONREPRESENTATION AS A RANDOM SUMSET OF NON-EXTINCTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 696 - 703
Copyright
Copyright © Applied Probability Trust 1980 

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References

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