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The multitype Galton-Watson process with ρ near 1

Published online by Cambridge University Press:  01 July 2016

M. P. Quine
M. P. Quine*
Affiliation:
University of Sydney
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Abstract

The asymptotic behaviour of those k-type (k ≥ 1) Galton-Watson processes (both with and without immigration) with ρ close to unity is considered. The first principal result (Theorem 3) relates to the vector denoting the numbers of particles of each type at the nth generation in processes without immigration. It is shown that, when normed in a certain way and conditioned on non-extinction, this vector has approximately, for large n, a negative exponential distribution which is degenerate on a line. Theorem 4 is an analogous result for processes with immigration. In this case, no conditioning is required, and the limiting distribution is again degenerate on a line, although now it relates to the gamma rather than the negative exponential.

Keywords

BRANCHING PROCESSMULTITYPEGALTON-WATSONPOSITIVELY REGULARNEAR CRITICALWITH IMMIGRATIONWITHOUT IMMIGRATION
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 3 , December 1972 , pp. 429 - 452
Copyright
Copyright © Applied Probability Trust 1972 

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