### Abstract

Consider a centred random walk in dimension one with a positive finite variance σ^{2}, and let τ_{B} be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic P_{x}(τ_{B}>n)∼2/πσ^{-1}V_{B}(x)n^{-1/2} and provide an explicit formula for the limit V_{B} as a function of the initial position x of the walk. We also give a functional limit theorem for the walk conditioned to avoid B by the time n. As a main application, we consider the case that B is an interval and study the size of the largest gap G_{n} (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for G_{n}, which is shown to be of the constant order, and describe its limit distribution. In addition, we prove an analogous result for the number of non-visited sites within the range of an integer-valued random walk.

Original language | English (US) |
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Pages (from-to) | 1886-1910 |

Number of pages | 25 |

Journal | Stochastic Processes and their Applications |

Volume | 125 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2015 |

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### Keywords

- Conditional limit theorem
- Harmonic function
- Hitting time
- Killed random walk
- Largest gap
- Limit theorem
- Maximal spacing
- Number of non-visited sites
- Random walk

### ASJC Scopus subject areas

- Modeling and Simulation
- Statistics and Probability
- Applied Mathematics