### Abstract

For each positive integer n ≥ 1, let Z_{2}^{n} be the direct product of n copies of Z_{2}, i.e., Z_{2}^{n} = {(a_{1}, a_{2}, ..., a_{n})∥a_{i} = 0 or 1 for all i = 1, 2, ..., n} and let {W_{t}^{n}}_{t≥0} be a random walk on Z_{2}^{n} such that P{W_{0}^{n} = A} = 2^{-n} for all A's in Z_{2}^{n} and P{W_{j+1}^{n} = (a_{2}, a_{3}, ..., a_{n}, 0)∥W_{j}^{n} = (a_{1}, a_{2}, ..., a_{n})} = P{W_{j+1}^{n} = (a_{2}, a_{3}, ..., a_{n}, 1)∥W_{j}^{n} = (a_{1}, a_{2}, ..., a_{n})} = 1 2 for all j = 0, 1, 2, ..., and all (a_{1}, a_{2}, ..., a_{n})'s in Z_{2}^{n}. For each positive integer n ≥ 1, let C_{n} denote the covering time taken by the random walk W_{t}^{n} on Z_{2}^{n} to cover Z_{2}^{n}, i.e., to visit every element of Z_{2}^{n}. In this paper, we prove that, among other results, P{except finitely many n, c2^{n}ln(2^{n}) < C_{n} < d2^{n}ln(2^{n})} = 1 if c < 1 < d.

Original language | English |
---|---|

Pages (from-to) | 111-118 |

Number of pages | 8 |

Journal | Journal of Multivariate Analysis |

Volume | 25 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 1988 |

### Fingerprint

### Keywords

- Borel-Cantelli lemma
- covering time
- random walks

### ASJC Scopus subject areas

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

### Cite this

_{2}

^{n}.

*Journal of Multivariate Analysis*,

*25*(1), 111-118. https://doi.org/10.1016/0047-259X(88)90156-X