Article
On the Discrepancy of Random Walks on the Circle
Authors
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Abstract
Let \( X_1, X_2, \ldots \) be i.i.d. absolutely continuous random variables, let \( S_k = \sum_{j=1}^{k} X_j \pmod 1 \) and let \( D_N^* \) denote the star discrepancy of the sequence \( (S_k)_{1 \leq k \leq N} \). We determine the limit distribution of \( \sqrt{N} D_N^* \) and the weak limit of the sequence \( \sqrt{N}(F_N(t) - t) \) in the Skorohod space \( D[0,1] \), where \( F_N(t) \) denotes the empirical distribution function of the sequence \( (S_k)_{1 \leq k \leq N} \).
Keywords
i.i.d. sums mod 1, empirical distribution, discrepancy, weak convergence.
Citation
Bazarova, A., Berkes, I., & Raseta, M. (2019). On the discrepancy of random walks on the circle. Uniform Distribution Theory, 14(2), 73–86. https://doi.org/10.2478/udt-2019-0015
A. Bazarova, I. Berkes and M. Raseta, “On the discrepancy of random walks on the circle,” Uniform Distribution Theory, vol. 14, no. 2, pp. 73–86, 2019, doi: 10.2478/udt-2019-0015.
Bazarova A, Berkes I, Raseta M. On the discrepancy of random walks on the circle. Uniform Distribution Theory. 2019;14(2):73–86. doi:10.2478/udt-2019-0015.
Bazarova, A., Berkes, I. and Raseta, M. (2019), ‘On the discrepancy of random walks on the circle’, Uniform Distribution Theory, 14(2), pp. 73–86. Available at: https://doi.org/10.2478/udt-2019-0015.
Bazarova, Alina, et al. “On the Discrepancy of Random Walks on the Circle.” Uniform Distribution Theory, vol. 14, no. 2, 2019, pp. 73–86. https://doi.org/10.2478/udt-2019-0015.
Bazarova, Alina, Istvan Berkes, and Marko Raseta. “On the Discrepancy of Random Walks on the Circle.” Uniform Distribution Theory 14, no. 2 (2019): 73–86. https://doi.org/10.2478/udt-2019-0015.
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Published by: Engineering Journals
