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A uniform limit theorem and exponential limit law for critical multitype age-dependent branching processes

Published online by Cambridge University Press:  14 July 2016

Martin I. Goldstein
Martin I. Goldstein*
Affiliation:
Université de Montréal
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Abstract

Let Z(t) ··· (Z1(t), …, Zk (t)) be an indecomposable critical k-type age-dependent branching process with generating function F(s, t). Denote the right and left eigenvalues of the mean matrix M by u and v respectively and suppose μ is the vector of mean lifetimes, i.e. Mu = u, vM = v.

It is shown that, under second moment assumptions, uniformly for s ∈ ([0, 1]k of the form s = 1 – cu, c a constant. Here vμ is the componentwise product of the vectors and Q[u] is a constant.

This result is then used to give a new proof of the exponential limit law.

Keywords

AGE-DEPENDENT BRANCHING PROCESSCRITICAL MULTITYPE BRANCHING PROCESSUNIFORM LIMIT THEOREMEXPONENTIAL LIMIT LAW
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 2 , June 1978 , pp. 235 - 242
Copyright
Copyright © Applied Probability Trust 1978 

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References

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