Biased random walks on a Galton-Watson tree with leaves
Biased random walks on a Galton-Watson tree with leaves
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DOI：
10.1214/10-aop620
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发表时间：
2007-11
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影响因子：
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通讯作者：
G. B. Arous;A. Fribergh;N. Gantert;A. Hammond
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作者：
G. B. Arous;A. Fribergh;N. Gantert;A. Hammond
我们考虑一个高尔顿-沃森树上的有偏随机漫步$X_n$，其叶子处于亚弹道状态。我们证明存在一个显式常数$\gamma= \gamma(\beta) \in (0,1)$，依赖于偏差$\beta$，使得$X_n$的阶为$n^{\gamma}$。用$\Delta_n$表示水平$n$的命中时间，证明$\Delta_n/n^{1/\gamma}$是紧的。此外，我们还证明$\Delta_n/n^{1/\gamma}$不收敛（至少对于$\beta$的大值）。我们证明了沿数列$n_{\lambda}(k)=\lfloor \lambda \beta^{\gamma k}\rfloor$, $\Delta_n/n^{1/\gamma}$收敛于某些无限可分定律。证明的关键工具是高尔顿-沃森树的经典哈里斯分解，再生时间的新变体以及对i.i.d重尾随机变量的三角形数组的仔细分析。
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.