Biased random walks on a Galton-Watson tree with leaves

Journal article

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.

Authors: Arous, G; Fribergh, A; Gantert, N; Hammond, A

• Journal: Annals of Probability
• Volume: 40
• Issue: 1
• Pages: 280-338
• Publication date: 2007-11-23
• DOI: 10.1214/10-AOP620
• ISSN: 0091-1798
• Language: English
• Keywords: math.PR