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Binary Trees, Exploration Processes, and an Extended Ray-Knight Theorem

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  04 February 2016

Mamadou Ba ,
Etienne Pardoux  and
Ahmadou Bamba Sow
Mamadou Ba*
Affiliation:
Aix-Marseille Université
Etienne Pardoux*
Affiliation:
Aix-Marseille Université
Ahmadou Bamba Sow*
Affiliation:
Université Gaston Berger
*
∗ Postal address: Centre de Mathématiques et d'Informatique, LATP-CNRS, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France.
∗ Postal address: Centre de Mathématiques et d'Informatique, LATP-CNRS, Aix-Marseille Université, 39 rue F. Joliot-Curie, 13453 Marseille cedex 13, France.
∗∗∗ Postal address: LERSTAD, Université Gaston Berger, BP 234, Saint-Louis, Senegal. Email address: ahmadou-bamba.sow@ugb.edu.sn
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Abstract

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We study the bijection between binary Galton-Watson trees in continuous time and their exploration process, both in the subcritical and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to ∞. We thus deduce Delmas' generalization of the second Ray-Knight theorem.

Keywords

Galton-Watson processFeller's branchingexploration processRay-Knight theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles
Secondary: 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 210 - 225
Copyright
© Applied Probability Trust 

References

Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 1–28.CrossRefGoogle Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Delmas, J.-F. (2008). Height process for super-critical continuous state branching process. Markov Process. Relat. Fields 14, 309–326.Google Scholar
Duquesne, T. and Le Gall, J.-F. (2002). Random trees, Lévy processes and spatial branching processes. Astérisque 281, 147pp.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.CrossRefGoogle Scholar
Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (IMA Vol. Math. Appl. 84), Springer, New York, pp. 111–126.CrossRefGoogle Scholar
Grimvall, A. (1974). On the convergence of sequences of branching processes. Ann. Prob. 2, 1027–1045.CrossRefGoogle Scholar
Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Prob. 38, 348–395.CrossRefGoogle Scholar
Le Gall, J.-F. (1989). Marches aléatoires, mouvement brownien et processus de branchement. In Séminaire de Probabilités XXIII (Lecture Notes Math. 1372), Springer, Berlin, pp. 258–274.CrossRefGoogle Scholar
Pitman, J. and Winkel, M. (2005). Growth of the Brownian forest. Ann. Prob. 33, 2188–2211.CrossRefGoogle Scholar
Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Commun. Pure Appl. Math. 24, 147–225.CrossRefGoogle Scholar