Random walks with badly approximable numbers
Probability
2007-05-23 v1 Number Theory
Abstract
Using 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.
Cite
@article{arxiv.math/0102206,
title = {Random walks with badly approximable numbers},
author = {Doug Hensley and Francis Edward Su},
journal= {arXiv preprint arXiv:math/0102206},
year = {2007}
}
Comments
7 pages; to appear in DIMACS volume "Unusual Applications of Number Theory"; related work at http://www.math.hmc.edu/~su/papers.html