## Abstract

We consider an infinite Galton-Watson tree Γ and label the vertices v with a collection of i.i.d. random variables (Y_{v})_{v∈Γ}. In the case where the upper tail of the distribution of Y_{v} is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables Y_{v} have finite exponential moments, the normalization in the definition of the speed depends on the distribution of Y_{v}. Interpreting the random variables Y_{v} as displacements of the offspring from the parent, (Y_{v})_{v∈Γ} describes a branching random walk. The result on the speed gives a limit theorem for the maximum of the branching random walk, that is, for the position of the rightmost particle. In our case, this maximum grows faster than linear in time.

Original language | English |
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Pages (from-to) | 1219-1229 |

Number of pages | 11 |

Journal | Annals of Probability |

Volume | 28 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2000 |

Externally published | Yes |

## Keywords

- Branching random walk
- Galton-Watson tree
- Semiexponential distributions
- Sums of i.i.d. random variables
- Tree-indexed random walk