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2001 Discrepancy Convergence for the Drunkard's Walk on the Sphere
Francis Su
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Electron. J. Probab. 6: 1-20 (2001). DOI: 10.1214/EJP.v6-75


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.


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Francis Su. "Discrepancy Convergence for the Drunkard's Walk on the Sphere." Electron. J. Probab. 6 1 - 20, 2001.


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

zbMATH: 0978.60011
MathSciNet: MR1816045
Digital Object Identifier: 10.1214/EJP.v6-75

Primary: 60B15
Secondary: 43A85

Keywords: Discrepancy , Gelfand pairs , homogeneous spaces , Legendre polynomials , Random walk

Vol.6 • 2001
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