New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices

According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy $$D_N^{*s}$$ of the first N rows and s columns, interpreted as N points in $$[0,1]^s$$ , satisfies an inequality of the form $$ D_N^{*s} \le \sqrt{\alpha } \sqrt{A+B\frac{\ln (\log _2(N))}{s}}\sqrt{\frac{s}{N}} $$ with $$\alpha = \zeta ^{-1}(2) \approx 1.73, A=1165$$ and $$B=178$$ . These matrices are obtained by using i.i.d sequences, and the parameters s and N refer to the dimension and the sample size respectively. In this paper, we improve their result in two directions: First, we change the character of the equation so that the constant A gets replaced by a value $$A_s$$ dependent on the dimension s such that for $$s>1$$ we have $$A_s