Zeno's walk: A random walk with refinements |
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Authors: | David Steinsaltz |
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Institution: | (1) Department of Mathematics, Harvard University, Cambridge, MA 02138, USA (e-mail: dstein@math.tu-berlin.de), US |
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Abstract: | Summary. A self-modifying random walk on 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 having Hausdorff dimension less than , 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.
Received: 20 November 1995 / In revised form: 14 May 1996 |
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Keywords: | Mathematics Subject Classification (1991):60J15 60F15 60F20 |
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