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Chen, Yuan-Hong; Wu, Jun
Stochastic processes and their applications, April 2020, 2020-04-00, Letnik: 130, Številka: 4Journal Article
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.
