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Elementary new proofs of classical limit theorems for Galton–Watson processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Jochen Geiger
Jochen Geiger*
Affiliation:
Universität Frankfurt
*
∗Postal address: Fachbereich Mathematik, Universität Frankfurt, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany.
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Abstract

Classical results describe the asymptotic behaviour of a Galton–Watson branching process conditioned on non-extinction. We give new proofs of limit theorems in critical and subcritical cases. The proofs are based on the representation of conditioned Galton–Watson generation sizes as a sum of independent increments which is derived from the decomposition of the conditioned Galton–Watson family tree along the line of descent of the left-most particle.

Keywords

Galton–Watson treebranching processconditioned limit laws

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 301 - 309
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by Deutsche Forschungsgemeinschaft.

References

Asmussen, S., and Hering, H. (1983). Branching Processes. Birkhäuser, Boston.CrossRefGoogle Scholar
Athreya, K. B., and Ney, P. (1972). Branching Processes. Springer, New York.Google Scholar
Bennies, J., and Kersting, G. (1999). A random walk approach to Galton–Watson trees. Preprint.Google Scholar
Chauvin, B., Rouault, A., and Wakolbinger, A. (1991). Growing conditioned trees. Stoch. Proc. Appl. 39, 117–130.CrossRefGoogle Scholar
Geiger, J. (1996). Size-biased and conditioned random splitting trees. Stoch. Proc. Appl. 65, 187–207.Google Scholar
Heathcote, C. R., Seneta, E., and Vere–Jones, D. (1967). A refinement of two theorems in the theory of branching processes. Theory Prob. Appl. 12, 297–301.Google Scholar
Joffe, A. (1967). On the Galton–Watson process with mean less than one. Ann. Math. Statist. 38, 264–266.Google Scholar
Kallenberg, O. (1977). Stability of critical cluster fields. Math. Nachr. 77, 7–43.Google Scholar
Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. Henri Poincaré Prob. Statist. 22, 425–487.Google Scholar
Kesten, H., Ney, P., and Spitzer, F. (1966). The Galton–Watson process with mean one and finite variance. Theory Prob. Appl. 11, 513–540.Google Scholar
Kolmogorov, A. N. (1938). Zur Lösung einer biologischen Aufgabe. Izv. NII Mathem. Mekh. Tomskogo Univ. 2, 1–6.Google Scholar
Lamperti, J., and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Theory Prob. Appl. 13, 128–139.CrossRefGoogle Scholar
Lyons, R., Pemantle, R., and Peres, Y. (1995). Conceptual proofs of L log L criteria for mean behavior of branching processes. Ann. Prob. 25, 1125–1138.Google Scholar
Yaglom, A. M. (1947). Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk. SSSR 56, 795–798.Google Scholar
Zubkov, A. M. (1975). Limiting distributions of the distance to the closest common ancestor. Theory Prob. Appl. 20, 602–612.Google Scholar