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Zeno's walk: A random walk with refinements
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  • Published: 01 October 2002

Zeno's walk: A random walk with refinements

  • David Steinsaltz1 

Probability Theory and Related Fields volume 107, pages 99–121 (1997)Cite this article

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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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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (e-mail: dstein@math.tu-berlin.de), Cambridge, MA, USA

    David Steinsaltz

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  1. David Steinsaltz
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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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  • Published: 01 October 2002

  • Issue Date: January 1997

  • DOI: https://doi.org/10.1007/s004400050078

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  • Mathematics Subject Classification (1991):60J15, 60F15, 60F20
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