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Labelling experiments and phase-age distributions for multiphase branching processes

Published online by Cambridge University Press:  14 July 2016

P. J. Brockwell  and
W. H. Kuo
P. J. Brockwell
Affiliation:
Michigan State University
W. H. Kuo
Affiliation:
Michigan State University
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Abstract

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

Keywords

AGE-DEPENDENT MULTIPHASE BRANCHING PROCESSASYMPTOTIC AGE DISTRIBUTIONMITOTIC CYCLECELL KINETICSLABELLING EXPERIMENT
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 4 , December 1973 , pp. 739 - 747
Copyright
Copyright © Applied Probability Trust 1973 

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