Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T15:20:36.012Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Alsmeyer, G. and Rösler, U. (2002). Asexual versus promiscuous bisexual Galton–Watson processes: the extinction probability ratio. Ann. Appl. Prob. 12, 125–142.Google Scholar
Alsmeyer, G. and Rösler, U. (2006). The Martin entrance boundary of the Galton–Watson process. Ann. Inst. H. Poincaré Prob. Statist. 42, 591–606.CrossRefGoogle Scholar
Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.Google Scholar
Asmussen, S. and Sigman, K. (1996). Monotone stochastic recursions and their duals. Prob. Eng. Inf. Sci. 10, 1–20.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.Google Scholar
Esty, W. W. (1975). The reverse Galton–Watson process. J. Appl. Prob. 12, 574–580.Google Scholar
Klebaner, F. C. and Sagitov, S. (2002). The age of a Galton–Watson population with a geometric offspring distribution. J. Appl. Prob. 39, 816–828.CrossRefGoogle Scholar
Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Theory Prob. Appl. 13, 128–139.Google Scholar
Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of Llog L criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.Google Scholar
Mitov, K. V., Pakes, A. G. and Yanev, G. P. (2003). Extremes of geometric variables with applications to branching processes. Statist. Prob. Lett. 65, 379–388.Google Scholar
Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales, Vol. 1, Foundations, 2nd edn. John Wiley, Chichester.Google Scholar
Spitzer, F. (1967). Two explicit Martin boundary constructions. In 1967 Symp. Prob. Methods Anal. (Lecture Notes Math. 31), Springer, Berlin, pp. 296–298.CrossRefGoogle Scholar