## Electronic Journal of Probability

### Discrepancy Convergence for the Drunkard's Walk on the Sphere

Francis Su

#### 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

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

Rights

#### 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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