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Pré-Publication, Document De Travail Année : 2008

Biased random walks on a Galton-Watson tree with leaves

Alexander Fribergh
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Nina Gantert
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Alan Hammond
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Résumé

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.
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Dates et versions

hal-00189984 , version 1 (22-11-2007)
hal-00189984 , version 2 (10-12-2007)
hal-00189984 , version 3 (20-10-2008)
hal-00189984 , version 4 (17-11-2010)

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Gérard Ben Arous, Alexander Fribergh, Nina Gantert, Alan Hammond. Biased random walks on a Galton-Watson tree with leaves. 2008. ⟨hal-00189984v3⟩

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