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Transformations of Galton-Watson processes and linear fractional reproduction

  • F. C. Klebaner (a1), U. Rösler (a2) and S. Sagitov (a3)

Abstract

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 N 2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

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Copyright

Corresponding author

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

Footnotes

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Supported by the Australian Research Council grant DP0451657.

Partially supported by the Bank of Sweden Tercentenary Foundation.

Footnotes

References

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[1] Alsmeyer, G. and Rösler, U. (2002). Asexual versus promiscuous bisexual Galton–Watson processes: the extinction probability ratio. Ann. Appl. Prob. 12, 125142.
[2] Alsmeyer, G. and Rösler, U. (2006). The Martin entrance boundary of the Galton–Watson process. Ann. Inst. H. Poincaré Prob. Statist. 42, 591606.
[3] Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhäuser, Boston, MA.
[4] Asmussen, S. and Sigman, K. (1996). Monotone stochastic recursions and their duals. Prob. Eng. Inf. Sci. 10, 120.
[5] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
[6] Esty, W. W. (1975). The reverse Galton–Watson process. J. Appl. Prob. 12, 574580.
[7] Klebaner, F. C. and Sagitov, S. (2002). The age of a Galton–Watson population with a geometric offspring distribution. J. Appl. Prob. 39, 816828.
[8] Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Theory Prob. Appl. 13, 128139.
[9] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of Llog L criteria for mean behavior of branching processes. Ann. Prob. 23, 11251138.
[10] Mitov, K. V., Pakes, A. G. and Yanev, G. P. (2003). Extremes of geometric variables with applications to branching processes. Statist. Prob. Lett. 65, 379388.
[11] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales, Vol. 1, Foundations, 2nd edn. John Wiley, Chichester.
[12] Spitzer, F. (1967). Two explicit Martin boundary constructions. In 1967 Symp. Prob. Methods Anal. (Lecture Notes Math. 31), Springer, Berlin, pp. 296298.

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Transformations of Galton-Watson processes and linear fractional reproduction

  • F. C. Klebaner (a1), U. Rösler (a2) and S. Sagitov (a3)

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