### Abstract

For a fixed set I of positive integers we consider the set of paths (p_{o},..., p_{k}) of arbitrary length satisfying p_{l}-p_{l-1}∈I for l=2,..., k and p_{0}=1, p_{k}=n. Equipping it with the uniform distribution, the random path length T_{n} is studied. Asymptotic expansions of the moments of T_{n} are derived and its asymptotic normality is proved. The step lengths p_{l}-p_{l-1} are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values of p_{0} and p_{k} are not specified. In the special case I={m+1, m+2,...} (for some fixed m∈ℕ) we derive the exact distribution of a random "m-gap" subset of {1,..., n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a random m-gap subset is also presented.

Original language | English |
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Pages (from-to) | 83-98 |

Number of pages | 16 |

Journal | Monatshefte für Mathematik |

Volume | 116 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1 1993 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Monatshefte für Mathematik*,

*116*(2), 83-98. https://doi.org/10.1007/BF01404004