### Abstract

Take a centered random walk Sn and consider the sequence of its partial sums A_{n} := ∑^{n}_{i=1} S_{i}. Suppose S_{1} is in the domain of normal attraction of an α-stable law with 1 < α ≤ 2. Assuming that S_{1} is either right-exponential (i.e. P(S_{1} < x|S_{1} > 0) = e ^{-ax} for some a > 0 and all x > 0) or right-continuous (skip free), we prove that P{A_{1} > 0, . . . , A_{N} > 0} ~ C_{α}N^{1/(2α)-1/2} as N → ∞, where C_{α} > 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Original language | English (US) |
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Pages (from-to) | 195-213 |

Number of pages | 19 |

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

Volume | 50 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2014 |

### Keywords

- Area of excursion
- Area of random walk
- Integrated random walk
- One-sided exit probability
- Persistence
- Sparre-Andersen theorem
- Stable excursion
- Unilateral small deviations

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Vysotsky, V. (2014). Positivity of integrated random walks.

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,*50*(1), 195-213. https://doi.org/10.1214/12-AIHP487