We analyze the drunkard's walk on the unit sphere with step size theta and
show that the walk converges in order constant/sin^2(theta) steps in the
discrepancy metric. This is an application of techniques we develop for
bounding the discrepancy of random walks on Gelfand pairs generated by
bi-invariant measures. In such cases, Fourier analysis on the acting group
admits tractable computations involving spherical functions. We advocate the
use of discrepancy as a metric on probabilities for state spaces with isometric
group actions.Comment: 20 pages; to appear in Electron. J. Probab.; related work at
  http://www.math.hmc.edu/~su/papers.htm

Su, Francis Edward

Electronic Journal of Probability

English

arXiv

. Fix an angle `, and consider the random walk on S  2  that starts at the north pole, and at each step moves by an angle ` in any uniformly chosen direction. We show that  C  sin 2 ` steps are necessary and sufficient to make the discrepancy distance from the uniform distribution small. Methods are derived for handling the discrepancy of random walks on arbitrary Gelfand pairs generated by bi-invariant measures. 1. Introduction  Consider the following random walk on the sphere S  2  . Fix an angle `,  measured from the center of the sphere. The random walk starts at the north pole, and at each step moves along the surface of the sphere by an angle ` in any uniformly chosen direction from the current position (hence mimicking the behavior of a &quot;drunkard&quot;). In this paper, we derive sharps rate of convergence for this walk under the discrepancy metric for measures on S  2  . This metric metrizes weak-* convergence of measures on S  2  . We show that  C  sin  2  `  steps are both necessar..

Francis Edward Su

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Discrepancy Convergence For The Drunkard&apos;s Walk On The Sphere

Francis Su

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Discrepancy Convergence for the Drunkard's Walk on the Sphere

We analyze the drunkard\u27s walk on the unit sphere with step size θ and show that the walk converges in order C/sin2(θ) steps in the discrepancy metric (C a constant). This is an application of techniques we develop for bounding the discrepancy of random walks on Gelfand pairs generated by bi-invariant measures. In such cases, Fourier analysis on the acting group admits tractable computations involving spherical functions. We advocate the use of discrepancy as a metric on probabilities for state spaces with isometric group actions

Su, Francis E.

Scholarship@Claremont

Discrepancy Convergence for the Drunkard\u27s Walk on the Sphere

Keck Graduate Institute

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.8829

Discrepancy convergence for the drunkard's walk on the sphere

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Scholarship@Claremont

Keck Graduate Institute