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Full-Text Articles in Mathematics
Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su
Random Walks On The Torus With Several Generators, Timothy Prescott '02, Francis E. Su
All HMC Faculty Publications and Research
Given n vectors {i} ∈ [0, 1)d, consider a random walk on the d-dimensional torus d = ℝd/ℤd generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*k) ≥ C1k−n/2, where C1 = C(n, d) is …
A Leveque-Type Lower Bound For Discrepancy, Francis E. Su
A Leveque-Type Lower Bound For Discrepancy, Francis E. Su
All HMC Faculty Publications and Research
A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on Rk / Zk. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.