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On the asymptotic behaviour of branching processes with infinite mean

Published online by Cambridge University Press:  01 July 2016

H.-J. Schuh  and
A. D. Barbour
H.-J. Schuh
Affiliation:
Division of Mathematics and Statistics, C.S.I.R.O., Canberra
A. D. Barbour
Affiliation:
University of Cambridge
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Abstract

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn}). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn}, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn} such that In Part 1 we study the properties of {Zn/Cn} and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn(Zn)}, where {yn} is a suitable chosen sequence of functions related to {Zn}. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn)/Cn}, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn} it is possible to construct explicitly functions U, such that U(Zn)/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.

Keywords

GALTON-WATSON PROCESS WITH INFINITE MEANALMOST SURE CONVERGENCEMARTINGALESNON-DEGENERATE AND PROPER RANDOM VARIABLECUMULANT GENERATING FUNCTIONREGULAR AND IRREGULAR POINTSREGULAR AND IRREGULAR PROCESSESSLOWLY VARYING FUNCTIONSPOINCARÉFUNCTIONAL EQUATION
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 4 , December 1977 , pp. 681 - 723
Copyright
Copyright © Applied Probability Trust 1977 

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