### 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 γ = γ(β) ∈ (0, 1), depending on the bias β, such that |X _{n}| is of order n ^{γ}. Denoting {increment} _{n} the hitting time of level n, we prove that {increment} _{n}/n ^{1/γ} is tight. Moreover, we show that {increment} _{n}/n ^{1/γ} does not converge in law (at least for large values of β). We prove that along the sequences n _{λ}(k) =⌊λβ ^{γk}⌋, {increment} _{n}/n ^{1/γ} 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.

Original language | English (US) |
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Pages (from-to) | 280-338 |

Number of pages | 59 |

Journal | Annals of Probability |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

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### Keywords

- Electrical networks
- Galton-Watson tree
- Infinitely divisible distributions
- Random walk in random environment

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*40*(1), 280-338. https://doi.org/10.1214/10-AOP620