Electronic Journal of Probability

Discrepancy Convergence for the Drunkard's Walk on the Sphere

Francis Su

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We analyze the drunkard's walk on the unit sphere with step size $\theta$ and show that the walk converges in order $C/\sin^2(\theta)$ 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.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 2, 20 pp.

Accepted: 19 February 2001
First available in Project Euclid: 19 April 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 43A85: Analysis on homogeneous spaces

discrepancy random walk Gelfand pairs homogeneous spaces Legendre polynomials

This work is licensed under aCreative Commons Attribution 3.0 License.


Su, Francis. Discrepancy Convergence for the Drunkard's Walk on the Sphere. Electron. J. Probab. 6 (2001), paper no. 2, 20 pp. doi:10.1214/EJP.v6-75. https://projecteuclid.org/euclid.ejp/1461097632

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