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Transformations of Galton-Watson processes and linear fractional reproduction

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

F. C. Klebaner ,
U. Rösler  and
S. Sagitov
F. C. Klebaner*
Affiliation:
Monash University
U. Rösler*
Affiliation:
Christian-Albrechts-Universität zu Kiel
S. Sagitov*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia. Email address: fima.klebaner@sci.monash.edu.au
∗∗ Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig Meyn Strasse 4, 24098 Kiel, Germany. Email address: roesler@math.uni-kiel.de
∗∗∗ Postal address: School of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden. Email address: serik@math.chalmers.se
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Abstract

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By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

Keywords

Galton-Watson processMarkov chain transformtime reversalDoob's transformcone duallinear fractional generating functionQ-process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 1036 - 1053
Copyright
Copyright © Applied Probability Trust 2007 

Footnotes

Supported by the Australian Research Council grant DP0451657.

Partially supported by the Bank of Sweden Tercentenary Foundation.

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