We analyze the drunkard's 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.
© 2001 Francis Su
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Francis Edward Su. Discrepancy convergence for the drunkard's walk on the sphere. Electron. J. Probab., 6:no. 2, 20 pp. (electronic), 2001.