Electronic Journal of Probability

Discrepancy Convergence for the Drunkard's Walk on the Sphere

Francis Su

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1461097632

Digital Object Identifier
doi:10.1214/EJP.v6-75

Mathematical Reviews number (MathSciNet)
MR1816045

Zentralblatt MATH identifier
0978.60011

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

Keywords
discrepancy random walk Gelfand pairs homogeneous spaces Legendre polynomials

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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