### 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 | |

State | Published - 2015 |

### Fingerprint

### 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

### Cite this

**Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem.** / Vysotsky, Vladislav.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 125, no. 5, pp. 1886-1910. https://doi.org/10.1016/j.spa.2014.11.017

}

TY - JOUR

T1 - Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem

AU - Vysotsky, Vladislav

PY - 2015

Y1 - 2015

N2 - Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τB>n)∼2/πσ-1VB(x)n-1/2 and provide an explicit formula for the limit VB 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 Gn (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for Gn, 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.

AB - Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τB>n)∼2/πσ-1VB(x)n-1/2 and provide an explicit formula for the limit VB 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 Gn (maximal spacing) within the range of the walk by the time n. We prove a limit theorem for Gn, 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.

KW - Conditional limit theorem

KW - Harmonic function

KW - Hitting time

KW - Killed random walk

KW - Largest gap

KW - Limit theorem

KW - Maximal spacing

KW - Number of non-visited sites

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=84923574658&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923574658&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2014.11.017

DO - 10.1016/j.spa.2014.11.017

M3 - Article

VL - 125

SP - 1886

EP - 1910

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 5

ER -