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Functional normalizations for the branching process with infinite mean

Published online by Cambridge University Press:  14 July 2016

Andrew D. Barbour  and
H.-J. Schuh
Andrew D. Barbour*
Affiliation:
Gonville and Caius College, Cambridge
H.-J. Schuh
Affiliation:
CSIRO Division of Mathematics and Statistics, Canberra
*
∗ Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, U.K.
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Abstract

It is well known that, in a Bienaymé-Galton–Watson process (Zn) with 1 < m = EZ1 < ∞ and EZ1 log Z1 <∞, the sequence of random variables Znm –n converges a.s. to a non–degenerate limit. When m =∞, an analogous result holds: for any 0< α < 1, it is possible to find functions U such that α n U (Zn) converges a.s. to a non-degenerate limit. In this paper, some sufficient conditions, expressed in terms of the probability generating function of Z1 and of its distribution function, are given under which a particular pair (α, U) is appropriate for (Zn). The most stringent set of conditions reduces, when U (x) x, to the requirements EZ1 = 1/α, EZ1 log Z1 <∞.

Keywords

BRANCHING PROCESSFUNCTIONAL NORMALIZATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 3 , September 1979 , pp. 513 - 525
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

∗∗

Present address: Department of Statistics, Richard Berry Building, University of Melbourne Parkville, Victoria 3052, Australia.

References

Davies, P. L. (1978) The simple branching process: a note on convergence when the mean is infinite. J. Appl. Prob. 15, 466–480.Google Scholar
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Kesten, H. and Stigum, B. P. (1966) A limit theorem for multi-dimensional Galton–Watson processes. Ann. Math. Statist. 37, 1211–1223.Google Scholar
Schuh, H-J. and Barbour, A. D. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.Google Scholar
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