### 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 - 1988 |

### Fingerprint

### Keywords

- Borel-Cantelli lemma
- covering time
- random walks

### ASJC Scopus subject areas

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

### Cite this

_{2}

^{n}.

*Journal of Multivariate Analysis*,

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

**Random walks on Z _{2}^{n}.** / Erdős, P.; Chen, Robert W.

Research output: Contribution to journal › Article

_{2}

^{n}',

*Journal of Multivariate Analysis*, vol. 25, no. 1, pp. 111-118. https://doi.org/10.1016/0047-259X(88)90156-X

_{2}

^{n}. Journal of Multivariate Analysis. 1988;25(1):111-118. https://doi.org/10.1016/0047-259X(88)90156-X

}

TY - JOUR

T1 - Random walks on Z2n

AU - Erdős, P.

AU - Chen, Robert W.

PY - 1988

Y1 - 1988

N2 - For each positive integer n ≥ 1, let Z2n be the direct product of n copies of Z2, i.e., Z2n = {(a1, a2, ..., an)∥ai = 0 or 1 for all i = 1, 2, ..., n} and let {Wtn}t≥0 be a random walk on Z2n such that P{W0n = A} = 2-n for all A's in Z2n and P{Wj+1n = (a2, a3, ..., an, 0)∥Wjn = (a1, a2, ..., an)} = P{Wj+1n = (a2, a3, ..., an, 1)∥Wjn = (a1, a2, ..., an)} = 1 2 for all j = 0, 1, 2, ..., and all (a1, a2, ..., an)'s in Z2n. For each positive integer n ≥ 1, let Cn denote the covering time taken by the random walk Wtn on Z2n to cover Z2n, i.e., to visit every element of Z2n. In this paper, we prove that, among other results, P{except finitely many n, c2nln(2n) <Cn nln(2n)} = 1 if c <1

AB - For each positive integer n ≥ 1, let Z2n be the direct product of n copies of Z2, i.e., Z2n = {(a1, a2, ..., an)∥ai = 0 or 1 for all i = 1, 2, ..., n} and let {Wtn}t≥0 be a random walk on Z2n such that P{W0n = A} = 2-n for all A's in Z2n and P{Wj+1n = (a2, a3, ..., an, 0)∥Wjn = (a1, a2, ..., an)} = P{Wj+1n = (a2, a3, ..., an, 1)∥Wjn = (a1, a2, ..., an)} = 1 2 for all j = 0, 1, 2, ..., and all (a1, a2, ..., an)'s in Z2n. For each positive integer n ≥ 1, let Cn denote the covering time taken by the random walk Wtn on Z2n to cover Z2n, i.e., to visit every element of Z2n. In this paper, we prove that, among other results, P{except finitely many n, c2nln(2n) <Cn nln(2n)} = 1 if c <1

KW - Borel-Cantelli lemma

KW - covering time

KW - random walks

UR - http://www.scopus.com/inward/record.url?scp=1542528305&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1542528305&partnerID=8YFLogxK

U2 - 10.1016/0047-259X(88)90156-X

DO - 10.1016/0047-259X(88)90156-X

M3 - Article

VL - 25

SP - 111

EP - 118

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -