## Abstract

We study a one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size X decays as P[X > t] ∼ a exp{−λt^{r} } for some constants a, λ > 0 where r ∈ (0, 1). We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost sure limit theorems, convergence in law and a growth condition dichotomy. The limit theorems reveal interesting differences betweens the two regimes r ∈ (0, 2/3) and r ∈ (2/3, 1), with yet different limits in the boundary case r = 2/3.

Original language | English |
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Pages (from-to) | 539-562 |

Number of pages | 24 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - May 2023 |

## Keywords

- Branching random walk
- Extreme values
- Limit theorems
- Point processes
- Stretched exponential random variables