Asymptotics of iterated branching processes
Piau Didier
Source: J. Appl. Probab.
Volume 46, Number 3
(2009), 917-924.
Abstract
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).
Primary Subjects: 60J80
Secondary Subjects: 92D10
Keywords: Branching process; trinucleotide repeat expansion; genetic disease
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jap/1253279859
Digital Object Identifier: doi:10.1239/jap/1253279859
Zentralblatt MATH identifier:
05611425
References
Gawe\l, B. and Kimmel, M. (1996). The iterated Galton--Watson process. J. Appl. Prob. 33, 949--959.
Kimmel, M. and Axelrod, D. E. (2002). Branching Processes in Biology (Interdisciplinary Appl. Math. 19). Springer, New York.
Ney, P. E. and Vidyashankar, A. N. (2003). Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Prob. 13, 475--489.
Pakes, A. G. (2003). Biological applications of branching processes. In Stochastic Processes: Modelling and Simulation (Handbook Statist. 21), North-Holland, Amsterdam, pp. 693--773.
Piau, D. (2004). Immortal branching Markov processes: averaging properties and PCR applications. Ann. Prob. 32, 337--364.
