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hal-00189984, version 4

## Biased random walks on a Galton-Watson tree with leaves

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

Abstract: 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

• hal-00189984, version 4
• oai:hal.archives-ouvertes.fr:hal-00189984
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• Submitted on: Wednesday, 17 November 2010 00:27:33
• Updated on: Wednesday, 17 November 2010 15:47:02