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Yule process approximation for the skeleton of a branching process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Neil O'Connell
Neil O'Connell*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Statistics, University of California, Berkeley, CA 94720, USA.
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Abstract

The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 + r, r ≧ 0, and finite offspring variance, is considered. When r = 0 it is trivial. If r > 0 is small and the time unit is taken as α /r generations (α > 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rate α. This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.

Keywords

DECOMPOSITION OF BRANCHING PROCESSESSUPERCRITICALPURE BIRTH PROCESSHARRIS–SEVAST'YANOV TRANSFORMATIONDIFFUSION APPROXIMATION

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 725 - 729
Copyright
Copyright © Applied Probability Trust 1993 

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References

Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
Eshel, I. (1981) On the survival probability of a slightly advantageous mutant gene with a general distribution of progeny size - a branching process model. J. Math. Biol. 12, 355–362.Google Scholar
Ethier, S. and Kurtz, T. (1986) Markov Processes: Characterisation and Convergence. Wiley, New York.Google Scholar
Evans, S. N. and O'Connell, N. (1993) Weighted occupation times for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. To appear.Google Scholar
Harris, T. (1948) Branching processes. Ann. Math. Statist. 19, 474–494.Google Scholar
Lamperti, J. and Ney, P. (1968) Conditioned branching processes and their limiting diffusions. Theory Prob. Appl. 13, 128–139.Google Scholar
Lindvall, T. (1974) Limit theorems for some functionals of certain Galton-Watson branching processes. Adv. Appl. Prob. 6, 309–321.Google Scholar
O'Connell, N. (1993) The Genealogy of Branching Processes. , University of California at Berkeley.Google Scholar