Summary.
A self-modifying random walk on \({\Bbb Q}\) is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of \({\Bbb R}\) having Hausdorff dimension less than \(1\), which we calculate by a theorem of Billingsley. By generating function techniques we then calculate the exponential rate of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the Wasserstein metric. We describe how the process may viewed as a random walk on the space of monotone piecewise linear functions, where moves are taken by successive compositions with a randomly chosen such function.
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Received: 20 November 1995 / In revised form: 14 May 1996
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Steinsaltz, D. Zeno's walk: A random walk with refinements. Probab Theory Relat Fields 107, 99–121 (1997). https://doi.org/10.1007/s004400050078
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DOI: https://doi.org/10.1007/s004400050078