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Supercritical age-dependent branching processes with immigration

Published online by Cambridge University Press:  14 July 2016

K. B. Athreya ,
P. R. Parthasarathy  and
G. Sankaranarayanan
K. B. Athreya
Affiliation:
Indian Institute of Science, Bangalore
P. R. Parthasarathy
Affiliation:
Annamalai University, Annamalai Nagar
G. Sankaranarayanan
Affiliation:
Annamalai University, Annamalai Nagar
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Abstract

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

Keywords

SUPERCRITICALIMMIGRATIONAGE-DEPENDENTBRANCHING PROCESSESMALTHUSIAN PARAMETERCONVERGENCE IN DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 4 , December 1974 , pp. 695 - 702
Copyright
Copyright © Applied Probability Trust 1974 

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References

[1]Athreya, K. B. (1969) On the super-critical one-dimensional age-dependent branching processes. Ann. Math. Statist. 40, 743–763.Google Scholar
[2]Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer Verlag, Berlin.Google Scholar
[3]Doney, R. A. (1971) The total progeny of branching processes. J. Appl. Prob. 8, 407–412.Google Scholar
[4]Levinson, N. (1960) Limiting theorems for age-dependent branching processes. Ill. J. Math. 4, 100–118.Google Scholar
[5]Parthasarathy, P. R. Age-dependent branching processes with random immigration. Submitted for publication.Google Scholar
[6]Sankaranarayanan, G. and Parthasarathy, P. R. (1972) Age-dependent branching processes, with immigration. Proceedings of the Indian Census Centenary Seminar, 1972.Google Scholar
[7]Seneta, E. (1970) On the super-critical Galton-Watson processes with immigration. Math. Biosci. 7, 9–14.Google Scholar