### Abstract

This paper is concerned with the small time behaviour of a Lévy process X. In particular, we investigate the stabilities of the times, T̄ _{b}(r) and T_{b}* (r), at which X, started with X _{0} = 0, first leaves the space-time regions {(t, y) ∈ ℝ^{2}: y ≤ rt^{b}, t ≥ 0} (one-sided exit), or {(t, y) ∈ ℝ^{2}: |y| ≤ rt^{b}, t ≥ 0} (two-sided exit), 0 ≤ b < 1, as r ↓ 0. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in L^{p}. In many instances these are seen to be equivalent to relative stability of the process X itself. The analogous large time problem is also discussed.

Original language | English (US) |
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Pages (from-to) | 208-235 |

Number of pages | 28 |

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

Volume | 49 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2013 |

### Keywords

- Lévy process
- Overshoot
- Passage times across power law boundaries
- Random walks
- Relative stability

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty