HAL : hal-00189984, version 4

arXiv : 0711.3686

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v1 (23-11-2007)

v2 (11-12-2007)

v3 (20-10-2008)

v4 (17-11-2010)

Biased random walks on a Galton-Watson tree with leaves

Gérard Ben Arous 1, Alexander Fribergh 1, Nina Gantert 2, Alan Hammond 3

(22/11/2007)

We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma= \gamma(\beta) \in (0,1)$, depending on the bias $\beta$, such that $X_n$ is of order $n^{\gamma}$. Denoting $\Delta_n$ the hitting time of level $n$, we prove that $\Delta_n/n^{1/\gamma}$ is tight. Moreover we show that $\Delta_n/n^{1/\gamma}$ does not converge in law (at least for large values of $\beta$). We prove that along the sequences $n_{\lambda}(k)=\lfloor \lambda \beta^{\gamma k}\rfloor$, $\Delta_n/n^{1/\gamma}$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d.~heavy-tailed random variables.

1 :	Courant Institute of Mathematical Science (CIMS)
	New York University
2 :	Institut für Mathematische Statistik
	Universitaet Muenster
3 :	Department of Statistics
	University of Oxford

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Mathématiques/Probabilités

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Contributeur : Alexander Fribergh <>

Soumis le : Mercredi 17 Novembre 2010, 00:27:33

Dernière modification le : Mercredi 17 Novembre 2010, 15:47:02