### Abstract

We study distributions F on [0, 8) such that for some T = 8, F*^{2}(x, x+T] ~ 2F(x, x+T]. The case T = 8 corresponds to F being subexponential, and our analysis shows that the properties for T < 8 are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman-Harris branching processes.

Original language | English |
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Pages (from-to) | 489-518 |

Number of pages | 30 |

Journal | Journal of Theoretical Probability |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2003 |

### Keywords

- Distribution tails
- Local probabilities
- Subexponential distributions
- Sums of independent random variables

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## Cite this

Asmussen, S., Foss, S., & Korshunov, D. (2003). Asymptotics for Sums of Random Variables with Local Subexponential Behaviour.

*Journal of Theoretical Probability*,*16*(2), 489-518. https://doi.org/10.1023/A:1023535030388