On the Discrepancy of Random Walks on the Circle

Let X₁,X₂,... be i.i.d. absolutely continuous random variables, let Sk=∑j=1kXj{S_k} = \sum\nolimits_{j = 1}^k {{X_j}} (mod 1) and let D*_N denote the star discrepancy of the sequence (S_k)_1≤k≤N. We determine the limit distribution of NDN*\sqrt N D_N^* and the weak limit of the sequence N(FN(t)-t)\sqrt N \left( {{F_N}(t) - t} \right) in the Skorohod space D[0, 1], where F_N (t) denotes the empirical distribution function of the sequence (S_k)_1≤k≤N.