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Heavy traffic approximations for the Galton-Watson process

Published online by Cambridge University Press:  01 July 2016

K. S. Fahady ,
M. P. Quine  and
D. Vere-Jones
K. S. Fahady
Affiliation:
University of Hull
M. P. Quine
Affiliation:
Australian National University
D. Vere-Jones
Affiliation:
Victoria University of Wellington
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Abstract

The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and conditional on non-extinction when there is no immigration, the number of individuals remaining in the population after a large number of generations has approximately a gamma distribution. The error estimates are uniform within a specified class of offspring distributions, and are independent of whether the critical situation is approached from above or below. These results parallel those given for continuous time branching processes by Sevast'yanov (1959), and extend recent work by Nagaev and Mohammedhanova (1966), Quineand Seneta (1969), and Seneta (1970).

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 3 , Issue 2 , Autumn 1971 , pp. 282 - 300
Copyright
Copyright © Applied Probability Trust 1971 

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