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Asymptotics of Iterated Branching Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Didier Piau
Didier Piau*
Affiliation:
Université Lyon 1
*
∗Current address: Institut Fourier UMR 5582, Université Joseph Fourier Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint Martin d'Hères, France. Email address: [email protected]
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Abstract

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Gaweł and Kimmel (1996) introduced and studied iterated Galton–Watson processes, (Xn)n≥0, possibly with thinning, as models of the number of repeats of DNA triplets during some genetic disorders. Our main results are the following. If the process indeed involves some thinning then extinction, {Xn→0}, and explosion, {Xn→∞}, can have positive probability simultaneously. If the underlying (simple) Galton–Watson process is nondecreasing with mean m then, conditionally on explosion, the ratios (log Xn+1)/Xn converge to logm almost surely. This simplifies the arguments of Gaweł and Kimmel, and confirms and extends a conjecture of Pakes (2003).

Keywords

Branching processtrinucleotide repeat expansiongenetic disease

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D10: Genetics
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 917 - 924
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Gaweł, B. and Kimmel, M. (1996). The iterated Galton–Watson process. J. Appl. Prob. 33, 949–959.CrossRefGoogle Scholar
[2] Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology (Interdisciplinary Appl. Math. 19). Springer, New York.Google Scholar
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[4] Pakes, A. G. (2003). Biological applications of branching processes. In Stochastic Processes: Modelling and Simulation (Handbook Statist. 21), North-Holland, Amsterdam, pp. 693–773.Google Scholar
[5] Piau, D. (2004). Immortal branching Markov processes: averaging properties and PCR applications. Ann. Prob. 32, 337–364.CrossRefGoogle Scholar