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29 March 2024
 
  » arxiv » math.PR/0102206

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Random walks with badly approximable numbers
Doug Hensley ; Francis Edward Su ;
Date 27 Feb 2001
Journal DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 64 (2004), 95-101.
Subject Probability; Number Theory MSC-class: 60B15 (Primary) 11J13, 11K38, 11K60 (Secondary) | math.PR math.NT
AbstractUsing the discrepancy metric, we analyze the rate of convergence of a random walk on the circle generated by d rotations, and establish sharp rates that show that badly approximable d-tuples in R^d give rise to walks with the fastest convergence. We use the discrepancy metric because the walk does not converge in total variation. For badly approximable d-tuples, the discrepancy is bounded above and below by (constant)k^(-d/2), where k is the number of steps in the random walk. We show how the constants depend on the d-tuple.
Source arXiv, math.PR/0102206
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