HAL : hal-00189984, version 4
 arXiv : 0711.3686
 Versions disponibles : 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
 (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
 Domaine : Mathématiques/Probabilités
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 hal-00189984, version 4 http://hal.archives-ouvertes.fr/hal-00189984 oai:hal.archives-ouvertes.fr:hal-00189984 Contributeur : Alexander Fribergh <> Soumis le : Mercredi 17 Novembre 2010, 00:27:33 Dernière modification le : Mercredi 17 Novembre 2010, 15:47:02