Random walks on Z2n

Paul Erdös, Robert W. Chen

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

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 < d2nln(2n)} = 1 if c < 1 < d.

Original languageEnglish
Pages (from-to)111-118
Number of pages8
JournalJournal of Multivariate Analysis
Volume25
Issue number1
DOIs
Publication statusPublished - Apr 1988

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Keywords

  • Borel-Cantelli lemma
  • covering time
  • random walks

ASJC Scopus subject areas

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

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