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On the range of simple symmetric random walks on the line

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  • Chen, Yuan-Hong
  • Wu, Jun

Abstract

This paper is aimed at a detailed study of the behaviors of random walks which is defined by the dyadic expansions of points. More precisely, let x=(ϵ1(x),ϵ2(x),…) be the dyadic expansion for a point x∈[0,1) and Sn(x)=∑k=1n(2ϵk(x)−1), which can be regarded as a simple symmetric random walk on Z. Denote by Rn(x) the cardinality of the set {S1(x),…,Sn(x)}, which is just the distinct position of x passed after n times. The set of points whose behavior satisfies Rn(x)∼cnγ is studied (c>0 and 0<γ≤1 being fixed) and its Hausdorff dimension is calculated.

Suggested Citation

  • Chen, Yuan-Hong & Wu, Jun, 2020. "On the range of simple symmetric random walks on the line," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2282-2295.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2282-2295
    DOI: 10.1016/j.spa.2019.07.004
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