Biased random walks on a Galton-Watson tree with leaves

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

1 CIMS - Courant Institute of Mathematical Sciences [New York]
2 Institut für Mathematische Statistik
3 Department of Statistics [Oxford]

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.

Domaines: Probabilités [math.PR]

Gérard Ben Arous, Alexander Fribergh, Nina Gantert, Alan Hammond. Biased random walks on a Galton-Watson tree with leaves. 2010. ⟨hal-00189984v4⟩